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Theorems that help decompose a finite group based on prime factors of its order
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.
For a prime number $p$, a Sylow "p"-subgroup (sometimes "p"-Sylow subgroup) of a group $G$ is a maximal $p$-subgroup of $G$, i.e., a subgroup of $G$ that is a "p"-group (meaning its cardinality is a power of $p,$ or equivalently, the order of every group element is a power of $p$) that is not a proper subgroup of any other $p$-subgroup of $G$. The set of all Sylow $p$-subgroups for a given prime $p$ is sometimes written $\text{Syl}_p(G)$.
The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group $G$ the order (number of elements) of every subgroup of $G$ divides the order of $G$. The Sylow theorems state that for every prime factor "$p$" of the order of a finite group $G$, there exists a Sylow $p$-subgroup of $G$ of order $p^n$, the highest power of $p$ that divides the order of $G$. Moreover, every subgroup of order "$p^n$" is a Sylow "$p$"-subgroup of $G$, and the Sylow $p$-subgroups of a group (for a given prime $p$) are conjugate to each other. Furthermore, the number of Sylow $p$-subgroups of a group for a given prime $p$ is congruent to 1 (mod $p$).
Theorems.
Motivation.
The Sylow theorems are a powerful statement about the structure of groups in general, but are also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group $G$ to give statements about the structure of its subgroups: essentially, it gives a technique to transport basic number-theoretic information about a group to its group structure. From this observation, classifying finite groups becomes a game of finding which combinations/constructions of groups of smaller order can be applied to construct a group. For example, a typical application of these theorems is in the classification of finite groups of some fixed cardinality, e.g. $|G| = 60$.
Statement.
Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of $\operatorname{Syl}_p(G)$, all members are actually isomorphic to each other and have the largest possible order: if $|G|=p^nm$ with $n > 0$ where p does not divide m, then every Sylow p-subgroup P has order $|P| = p^n$. That is, P is a p-group and $\text{gcd}(|G:P|, p) = 1$. These properties can be exploited to further analyze the structure of G.
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in "Mathematische Annalen".
The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy, and is known as Cauchy's theorem.
Theorem (3) — G:N_G(P)
Consequences.
The Sylow theorems imply that for a prime number $p$ every Sylow $p$-subgroup is of the same order, $p^n$. Conversely, if a subgroup has order $p^n$, then it is a Sylow $p$-subgroup, and so is conjugate to every other Sylow $p$-subgroup. Due to the maximality condition, if $H$ is any $p$-subgroup of $G$, then $H$ is a subgroup of a $p$-subgroup of order $p^n$.
A very important consequence of Theorem 2 is that the condition $n_p = 1$ is equivalent to saying that the Sylow $p$-subgroup of $G$ is a normal subgroup. However, there are groups that have normal subgroups but no normal Sylow subgroups, such as $S_4$.
Sylow theorems for infinite groups.
There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow p-subgroup in an infinite group to be a "p"-subgroup (that is, every element in it has p-power order) that is maximal for inclusion among all p-subgroups in the group. Let $\operatorname{Cl}(K)$ denote the set of conjugates of a subgroup $K \subset G$.
Theorem — \operatorname{Cl}(K)
Examples.
A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the "n"-gon, "D"2"n". For "n" odd, 2 = 21 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are "n", and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.
By contrast, if "n" is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an outer automorphism, which can be represented by rotation through π/"n", half the minimal rotation in the dihedral group.
Another example are the Sylow p-subgroups of "GL"2("F""q"), where "p" and "q" are primes ≥ 3 and "p" ≡ 1 (mod "q") , which are all abelian. The order of "GL"2("F""q") is ("q"2 − 1)("q"2 − "q") = ("q")("q" + 1)("q" − 1)2. Since "q" = "p""n""m" + 1, the order of "GL"2("F""q") = "p"2"n" "m"′. Thus by Theorem 1, the order of the Sylow "p"-subgroups is "p"2"n".
One such subgroup "P", is the set of diagonal matrices $\begin{bmatrix}x^{im} & 0 \\0 & x^{jm} \end{bmatrix}$, "x" is any primitive root of "F""q". Since the order of "F""q" is "q" − 1, its primitive roots have order "q" − 1, which implies that "x"("q" − 1)/"p""n" or "x""m" and all its powers have an order which is a power of "p". So, "P" is a subgroup where all its elements have orders which are powers of "p". There are "pn" choices for both "a" and "b", making |"P"| = "p"2"n". This means "P" is a Sylow "p"-subgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow "p"-subgroups are conjugate to each other, the Sylow "p"-subgroups of "GL"2("F""q") are all abelian.
Example applications.
Since Sylow's theorem ensures the existence of p-subgroups of a finite group, it's worthwhile to study groups of prime power order more closely. Most of the examples use Sylow's theorem to prove that a group of a particular order is not simple. For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a normal subgroup.
Cyclic group orders.
Some non-prime numbers "n" are such that every group of order "n" is cyclic. One can show that "n" = 15 is such a number using the Sylow theorems: Let "G" be a group of order 15 = 3 · 5 and "n"3 be the number of Sylow 3-subgroups. Then "n"3 $\mid$ 5 and "n"3 ≡ 1 (mod 3). The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be normal (since it has no distinct conjugates). Similarly, "n"5 must divide 3, and "n"5 must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are coprime, the intersection of these two subgroups is trivial, and so "G" must be the internal direct product of groups of order 3 and 5, that is the cyclic group of order 15. Thus, there is only one group of order 15 (up to isomorphism).
Small groups are not simple.
A more complex example involves the order of the smallest simple group that is not cyclic. Burnside's "pa qb" theorem states that if the order of a group is the product of one or two prime powers, then it is solvable, and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30 (
2 · 3 · 5).
If "G" is simple, and |"G"| = 30, then "n"3 must divide 10 ( = 2 · 5), and "n"3 must equal 1 (mod 3). Therefore, "n"3 = 10, since neither 4 nor 7 divides 10, and if "n"3 = 1 then, as above, "G" would have a normal subgroup of order 3, and could not be simple. "G" then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus the identity). This means "G" has at least 20 distinct elements of order 3.
As well, "n"5 = 6, since "n"5 must divide 6 ( = 2 · 3), and "n"5 must equal 1 (mod 5). So "G" also has 24 distinct elements of order 5. But the order of "G" is only 30, so a simple group of order 30 cannot exist.
Next, suppose |"G"| = 42 = 2 · 3 · 7. Here "n"7 must divide 6 ( = 2 · 3) and "n"7 must equal 1 (mod 7), so "n"7 = 1. So, as before, "G" can not be simple.
On the other hand, for |"G"| = 60 = 22 · 3 · 5, then "n"3 = 10 and "n"5 = 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is "A"5, the alternating group over 5 elements. It has order 60, and has 24 cyclic permutations of order 5, and 20 of order 3.
Wilson's theorem.
Part of Wilson's theorem states that
$(p-1)! \equiv -1 \pmod p$
for every prime "p". One may easily prove this theorem by Sylow's third theorem. Indeed,
observe that the number "np" of Sylow's "p"-subgroups
in the symmetric group "Sp" is ("p" − 2)!. On the other hand, "n""p" ≡ 1 (mod "p"). Hence, ("p" − 2)! ≡ 1 (mod "p"). So, ("p" − 1)! ≡ −1 (mod "p").
Fusion results.
Frattini's argument shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. A slight generalization known as Burnside's fusion theorem states that if "G" is a finite group with Sylow "p"-subgroup "P" and two subsets "A" and "B" normalized by "P", then "A" and "B" are "G"-conjugate if and only if they are "NG"("P")-conjugate. The proof is a simple application of Sylow's theorem: If "B"="Ag", then the normalizer of "B" contains not only "P" but also "Pg" (since "Pg" is contained in the normalizer of "Ag"). By Sylow's theorem "P" and "Pg" are conjugate not only in "G", but in the normalizer of "B". Hence "gh"−1 normalizes "P" for some "h" that normalizes "B", and then "A""gh"−1 = "B"h−1 = "B", so that "A" and "B" are "NG"("P")-conjugate. Burnside's fusion theorem can be used to give a more powerful factorization called a semidirect product: if "G" is a finite group whose Sylow "p"-subgroup "P" is contained in the center of its normalizer, then "G" has a normal subgroup "K" of order coprime to "P", "G" = "PK" and "P"∩"K" = {1}, that is, "G" is "p"-nilpotent.
Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control a Sylow "p"-subgroup of the derived subgroup has on the structure of the entire group. This control is exploited at several stages of the classification of finite simple groups, and for instance defines the case divisions used in the Alperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedral group. These rely on J. L. Alperin's strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.
Proof of the Sylow theorems.
The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves is the subject of many papers, including Waterhouse, Scharlau, Casadio and Zappa, Gow, and to some extent Meo.
One proof of the Sylow theorems exploits the notion of group action in various creative ways. The group G acts on itself or on the set of its "p"-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems. The following proofs are based on combinatorial arguments of Wielandt. In the following, we use $a \mid b$ as notation for "a divides b" and $a \nmid b$ for the negation of this statement.
G (1) — $ is divisible by a prime power "pk" has a subgroup of order "pk".
Proof
\Omega
and no power of "p" remains in any of the factors inside the product on the right. Hence "νp"(|Ω|) = "νp"("m") = "r", completing the proof.
Lemma — Let H be a finite p-group, let Ω be a finite set acted on by H, and let Ω0 denote the set of points of Ω that are fixed under the action of H. Then |Ω| ≡ |Ω0| (mod p).
Proof
Any element x ∈ Ω not fixed by H will lie in an orbit of order | · |/|"Hx"| (where "Hx" denotes the stabilizer), which is a multiple of p by assumption. The result follows immediately by writing |Ω| as the sum of | · | over all distinct orbits Hx and reducing mod p.
Theorem (2) — If "H" is a "p"-subgroup of G and "P" is a Sylow "p"-subgroup of G, then there exists an element "g" in G such that "g"−1"Hg" ≤ "P". In particular, all Sylow "p"-subgroups of G are conjugate to each other (and therefore isomorphic), that is, if "H" and "K" are Sylow "p"-subgroups of G, then there exists an element "g" in G with "g"−1"Hg" = "K".
Proof
\Omega_0
Theorem (3) — Let "q" denote the order of any Sylow "p"-subgroup "P" of a finite group G. Let "np" denote the number of Sylow "p"-subgroups of G. Then (a) "np" = [G : "NG"("P")] (where "NG"("P") is the normalizer of "P"), (b) "np" divides | · |/"q", and (c) "np" ≡ 1 (mod "p").
Proof
Let Ω be the set of all Sylow "p"-subgroups of G and let G act on Ω by conjugation. Let "P" ∈ Ω be a Sylow "p"-subgroup. By Theorem 2, the orbit of "P" has size "np", so by the orbit-stabilizer theorem "np" = [G : G"P"]. For this group action, the stabilizer G"P" is given by {g ∈ G | "gPg"−1 = "P"} = "N"G("P"), the normalizer of "P" in G. Thus, "np" = [G : "NG"("P")], and it follows that this number is a divisor of [G : "P"] = | · |/"q".
Now let "P" act on Ω by conjugation, and again let Ω0 denote the set of fixed points of this action. Let "Q" ∈ Ω0 and observe that then "Q" = "xQx"−1 for all "x" ∈ "P" so that "P" ≤ "NG"("Q"). By Theorem 2, "P" and "Q" are conjugate in "NG"("Q") in particular, and "Q" is normal in "NG"("Q"), so then "P" = "Q". It follows that Ω0 = {"P"} so that, by the Lemma, |Ω| ≡ |Ω0| = 1 (mod "p").
Algorithms.
The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory.
One proof of the existence of Sylow "p"-subgroups is constructive: if "H" is a "p"-subgroup of "G" and the index ["G":"H"] is divisible by "p", then the normalizer "N" = "NG"("H") of "H" in "G" is also such that ["N" : "H"] is divisible by "p". In other words, a polycyclic generating system of a Sylow "p"-subgroup can be found by starting from any "p"-subgroup "H" (including the identity) and taking elements of "p"-power order contained in the normalizer of "H" but not in "H" itself. The algorithmic version of this (and many improvements) is described in textbook form in Butler, including the algorithm described in Cannon. These versions are still used in the GAP computer algebra system.
In permutation groups, it has been proven, in Kantor and Kantor and Taylor, that a Sylow "p"-subgroup and its normalizer can be found in polynomial time of the input (the degree of the group times the number of generators). These algorithms are described in textbook form in Seress, and are now becoming practical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the Magma computer algebra system.
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Group of even permutations of a finite set
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by A"n" or Alt("n").
Basic properties.
For "n" > 1, the group A"n" is the commutator subgroup of the symmetric group S"n" with index 2 and has therefore "n"!/2 elements. It is the kernel of the signature group homomorphism sgn : S"n" → {1, −1} explained under symmetric group.
The group A"n" is abelian if and only if "n" ≤ 3 and simple if and only if "n" = 3 or "n" ≥ 5. A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.
The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions {(), (12)(34), (13)(24), (14)(23)}, that is the kernel of the surjection of A4 onto A3 ≅ Z3. We have the exact sequence V → A4 → A3 = Z3. In Galois theory, this map, or rather the corresponding map S4 → S3, corresponds to associating the Lagrange resolvent cubic to a quartic, which allows the quartic polynomial to be solved by radicals, as established by Lodovico Ferrari.
Conjugacy classes.
As in the symmetric group, any two elements of A"n" that are conjugate by an element of A"n" must have the same cycle shape. The converse is not necessarily true, however. If the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type, then there are exactly two conjugacy classes for this cycle shape .
Examples:
"See Symmetric group".
Relation with symmetric group.
As finite symmetric groups are the groups of all permutations of a set with finite elements, and the alternating groups are groups of even permutations, alternating groups are subgroups of finite symmetric groups.
Generators and relations.
For "n" ≥ 3, A"n" is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions. This generating set is often used to prove that A"n" is simple for "n" ≥ 5.
Automorphism group.
For "n" > 3, except for "n" = 6, the automorphism group of A"n" is the symmetric group S"n", with inner automorphism group A"n" and outer automorphism group Z2; the outer automorphism comes from conjugation by an odd permutation.
For "n" = 1 and 2, the automorphism group is trivial. For "n" = 3 the automorphism group is Z2, with trivial inner automorphism group and outer automorphism group Z2.
The outer automorphism group of A6 is the Klein four-group V = Z2 × Z2, and is related to the outer automorphism of S6. The extra outer automorphism in A6 swaps the 3-cycles (like (123)) with elements of shape 32 (like (123)(456)).
Exceptional isomorphisms.
There are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are:
More obviously, A3 is isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group (which is also SL1("q") = PSL1("q") for any "q").
Example A5 as a subgroup of 3-space rotations.
A5 is the group of isometries of a dodecahedron in 3-space, so there is a representation A5 → SO3(R).
In this picture the vertices of the polyhedra represent the elements of the group, with the center of the sphere representing the identity element. Each vertex represents a rotation about the axis pointing from the center to that vertex, by an angle equal to the distance from the origin, in radians. Vertices in the same polyhedron are in the same conjugacy class. Since the conjugacy class equation for A5 is 1 + 12 + 12 + 15 + 20 = 60, we obtain four distinct (nontrivial) polyhedra.
The vertices of each polyhedron are in bijective correspondence with the elements of its conjugacy class, with the exception of the conjugacy class of (2,2)-cycles, which is represented by an icosidodecahedron on the outer surface, with its antipodal vertices identified with each other. The reason for this redundancy is that the corresponding rotations are by π radians, and so can be represented by a vector of length π in either of two directions. Thus the class of (2,2)-cycles contains 15 elements, while the icosidodecahedron has 30 vertices.
The two conjugacy classes of twelve 5-cycles in A5 are represented by two icosahedra, of radii 2π/5 and 4π/5, respectively. The nontrivial outer automorphism in Out(A5) ≃ Z2 interchanges these two classes and the corresponding icosahedra.
Example: the 15 puzzle.
It can be proved that the 15 puzzle, a famous example of the sliding puzzle, can be represented by the alternating group A15, because the combinations of the 15 puzzle can be generated by 3-cycles. In fact, any 2"k" − 1 sliding puzzle with square tiles of equal size can be represented by A2"k"−1.
Subgroups.
A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group "G" and a divisor "d" of |"G"|, there does not necessarily exist a subgroup of "G" with order "d": the group "G" = A4, of order 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any distinct nontrivial element generates the whole group.
For all "n" > 4, A"n" has no nontrivial (that is, proper) normal subgroups. Thus, A"n" is a simple group for all "n" > 4. A5 is the smallest non-solvable group.
Group homology.
The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sufficiently large "n", it is constant. However, there are some low-dimensional exceptional homology. Note that the homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions (additional homology elements).
"H"1: Abelianization.
The first homology group coincides with abelianization, and (since A"n" is perfect, except for the cited exceptions) is thus:
"H"1(A"n", Z) = Z1 for "n" = 0, 1, 2;
"H"1(A3, Z) = A = A3 = Z3;
"H"1(A4, Z) = A = Z3;
"H"1(A"n", Z) = Z1 for "n" ≥ 5.
This is easily seen directly, as follows. A"n" is generated by 3-cycles – so the only non-trivial abelianization maps are A"n" → Z3, since order-3 elements must map to order-3 elements – and for "n" ≥ 5 all 3-cycles are conjugate, so they must map to the same element in the abelianization, since conjugation is trivial in abelian groups. Thus a 3-cycle like (123) must map to the same element as its inverse (321), but thus must map to the identity, as it must then have order dividing 2 and 3, so the abelianization is trivial.
For "n" < 3, A"n" is trivial, and thus has trivial abelianization. For A3 and A4 one can compute the abelianization directly, noting that the 3-cycles form two conjugacy classes (rather than all being conjugate) and there are non-trivial maps A3 ↠ Z3 (in fact an isomorphism) and A4 ↠ Z3.
"H"2: Schur multipliers.
The Schur multipliers of the alternating groups A"n" (in the case where "n" is at least 5) are the cyclic groups of order 2, except in the case where "n" is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is (the cyclic group) of order 6. These were first computed in .
"H"2(A"n", Z) = Z1 for "n" = 1, 2, 3;
"H"2(A"n", Z) = Z2 for "n" = 4, 5;
"H"2(A"n", Z) = Z6 for "n" = 6, 7;
"H"2(A"n", Z) = Z2 for "n" ≥ 8.
Notes.
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Type of group in abstract algebra
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group $\mathrm{S}_n$ defined over a finite set of $n$ symbols consists of the permutations that can be performed on the $n$ symbols. Since there are $n!$ ($n$ factorial) such permutation operations, the order (number of elements) of the symmetric group $\mathrm{S}_n$ is $n!$.
Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.
The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. Cayley's theorem states that every group $G$ is isomorphic to a subgroup of the symmetric group on (the underlying set of) $G$.
Definition and first properties.
The symmetric group on a finite set $X$ is the group whose elements are all bijective functions from $X$ to $X$ and whose group operation is that of function composition. For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree $n$ is the symmetric group on the set $X = \{1, 2, \ldots, n\}$.
The symmetric group on a set $X$ is denoted in various ways, including $\mathrm{S}_X$, $\mathfrak{S}_X$, $\Sigma_X$, $X!$, and $\operatorname{Sym}(X)$. If $X$ is the set $\{1, 2, \ldots, n\}$ then the name may be abbreviated to $\mathrm{S}_n$, $\mathfrak{S}_n$, $\Sigma_n$, or $\operatorname{Sym}(n)$.
Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in , , and .
The symmetric group on a set of $n$ elements has order $n!$ (the factorial of $n$). It is abelian if and only if $n$ is less than or equal to 2. For $n=0$ and $n=1$ (the empty set and the singleton set), the symmetric groups are trivial (they have order $0! = 1! = 1$). The group S"n" is solvable if and only if $n \leq 4$. This is an essential part of the proof of the Abel–Ruffini theorem that shows that for every $n > 4$ there are polynomials of degree $n$ which are not solvable by radicals, that is, the solutions cannot be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the polynomial's coefficients.
Applications.
The symmetric group on a set of size "n" is the Galois group of the general polynomial of degree "n" and plays an important role in Galois theory. In invariant theory, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric functions. In the representation theory of Lie groups, the representation theory of the symmetric group plays a fundamental role through the ideas of Schur functors.
In the theory of Coxeter groups, the symmetric group is the Coxeter group of type A"n" and occurs as the Weyl group of the general linear group. In combinatorics, the symmetric groups, their elements (permutations), and their representations provide a rich source of problems involving Young tableaux, plactic monoids, and the Bruhat order. Subgroups of symmetric groups are called permutation groups and are widely studied because of their importance in understanding group actions, homogeneous spaces, and automorphism groups of graphs, such as the Higman–Sims group and the Higman–Sims graph.
Group properties and special elements.
The elements of the symmetric group on a set "X" are the permutations of "X".
Multiplication.
The group operation in a symmetric group is function composition, denoted by the symbol ∘ or simply by just a composition of the permutations. The composition "f" ∘ "g" of permutations "f" and "g", pronounced ""f" of "g"", maps any element "x" of "X" to "f"("g"("x")). Concretely, let (see permutation for an explanation of notation):
$ f = (1\ 3)(4\ 5)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end{pmatrix} $
$ g = (1\ 2\ 5)(3\ 4)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix}.$
Applying "f" after "g" maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing "f" and "g" gives
$ fg = f\circ g = (1\ 2\ 4)(3\ 5)=\begin{pmatrix} 1 & 2 &3 & 4 & 5 \\ 2 & 4 & 5 & 1 & 3\end{pmatrix}.$
A cycle of length "L" = "k" · "m", taken to the "k"th power, will decompose into "k" cycles of length "m": For example, ("k" = 2, "m" = 3),
$ (1~2~3~4~5~6)^2 = (1~3~5) (2~4~6).$
Verification of group axioms.
To check that the symmetric group on a set "X" is indeed a group, it is necessary to verify the group axioms of closure, associativity, identity, and inverses.
Transpositions, sign, and the alternating group.
A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation "g" from above can be written as "g" = (1 2)(2 5)(3 4). Since "g" can be written as a product of an odd number of transpositions, it is then called an odd permutation, whereas "f" is an even permutation.
The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. There are several short proofs of the invariance of this parity of a permutation.
The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd. Thus we can define the sign of a permutation:
$\operatorname{sgn}f = \begin{cases} +1, & \text{if }f\mbox { is even} \\ -1, & \text{if }f \text{ is odd}. \end{cases}$
With this definition,
$\operatorname{sgn}\colon \mathrm{S}_n \rightarrow \{+1, -1\}\ $
is a group homomorphism ({+1, −1} is a group under multiplication, where +1 is e, the neutral element). The kernel of this homomorphism, that is, the set of all even permutations, is called the alternating group A"n". It is a normal subgroup of S"n", and for "n" ≥ 2 it has "n"!/2 elements. The group S"n" is the semidirect product of A"n" and any subgroup generated by a single transposition.
Furthermore, every permutation can be written as a product of "adjacent transpositions", that is, transpositions of the form ("a" "a"+1). For instance, the permutation "g" from above can also be written as "g" = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5). The sorting algorithm bubble sort is an application of this fact. The representation of a permutation as a product of adjacent transpositions is also not unique.
Cycles.
A cycle of "length" "k" is a permutation "f" for which there exists an element "x" in {1, ..., "n"} such that "x", "f"("x"), "f"2("x"), ..., "f""k"("x") = "x" are the only elements moved by "f"; it conventionally is required that "k" ≥ 2 since with "k" = 1 the element "x" itself would not be moved either. The permutation "h" defined by
$h = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 1 & 3 & 5\end{pmatrix}$
is a cycle of length three, since "h"(1) = 4, "h"(4) = 3 and "h"(3) = 1, leaving 2 and 5 untouched. We denote such a cycle by (1 4 3), but it could equally well be written (4 3 1) or (3 1 4) by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are "disjoint" if they have disjoint subsets of elements. Disjoint cycles commute: for example, in S6 there is the equality (4 1 3)(2 5 6) = (2 5 6)(4 1 3). Every element of S"n" can be written as a product of disjoint cycles; this representation is unique up to the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point.
Cycles admit the following conjugation property with any permutation $\sigma$, this property is often used to obtain its generators and relations.
$\sigma\begin{pmatrix} a & b & c & \ldots \end{pmatrix}\sigma^{-1}=\begin{pmatrix}\sigma(a) & \sigma(b) & \sigma(c) & \ldots\end{pmatrix}$
Special elements.
Certain elements of the symmetric group of {1, 2, ..., "n"} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set).
The order reversing permutation is the one given by:
$\begin{pmatrix} 1 & 2 & \cdots & n\\
n & n-1 & \cdots & 1\end{pmatrix}.$
This is the unique maximal element with respect to the Bruhat order and the
longest element in the symmetric group with respect to generating set consisting of the adjacent transpositions ("i" "i"+1), 1 ≤ "i" ≤ "n" − 1.
This is an involution, and consists of $\lfloor n/2 \rfloor$ (non-adjacent) transpositions
$(1\,n)(2\,n-1)\cdots,\text{ or }\sum_{k=1}^{n-1} k = \frac{n(n-1)}{2}\text{ adjacent transpositions: }$
$(n\,n-1)(n-1\,n-2)\cdots(2\,1)(n-1\,n-2)(n-2\,n-3)\cdots,$
so it thus has sign:
$\mathrm{sgn}(\rho_n) = (-1)^{\lfloor n/2 \rfloor} =(-1)^{n(n-1)/2} = \begin{cases}
+1 & n \equiv 0,1 \pmod{4}\\
\end{cases}$
which is 4-periodic in "n".
In S2"n", the "perfect shuffle" is the permutation that splits the set into 2 piles and interleaves them. Its sign is also $(-1)^{\lfloor n/2 \rfloor}.$
Note that the reverse on "n" elements and perfect shuffle on 2"n" elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic.
Conjugacy classes.
The conjugacy classes of S"n" correspond to the cycle types of permutations; that is, two elements of S"n" are conjugate in S"n" if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S5, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of S"n" can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Continuing the previous example,
<math display="block">k = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 3 & 2 & 5\end{pmatrix},$
which can be written as the product of cycles as (2 4).
This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is,
<math display="block">(2~4)\circ(1~2~3)(4~5)\circ(2~4)=(1~4~3)(2~5).$
It is clear that such a permutation is not unique.
Conjugacy classes of S"n" correspond to integer partitions of "n": to the partition "μ" = ("μ"1, "μ"2, ..., "μ""k") with <math display="inline">n=\sum_{i=1}^k \mu_i$ and "μ"1 ≥ "μ"2 ≥ ... ≥ "μ""k", is associated the set "C""μ" of permutations with cycles of lengths "μ"1, "μ"2, ..., "μ""k". Then "C""μ" is a conjugacy class of S"n", whose elements are said to be of cycle-type $\mu$.
Low degree groups.
The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.
Note that while A6 and A7 have an exceptional Schur multiplier (a triple cover) and that these extend to triple covers of S6 and S7, these do not correspond to exceptional Schur multipliers of the symmetric group.
Maps between symmetric groups.
Other than the trivial map S"n" → C1 ≅ S0 ≅ S1 and the sign map S"n" → S2, the most notable homomorphisms between symmetric groups, in order of relative dimension, are:
There are also a host of other homomorphisms S"m" → S"n" where "m" < "n".
Relation with alternating group.
For "n" ≥ 5, the alternating group A"n" is simple, and the induced quotient is the sign map: A"n" → S"n" → S2 which is split by taking a transposition of two elements. Thus S"n" is the semidirect product A"n" ⋊ S2, and has no other proper normal subgroups, as they would intersect A"n" in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in A"n" (and thus themselves be A"n" or S"n").
S"n" acts on its subgroup A"n" by conjugation, and for "n" ≠ 6, S"n" is the full automorphism group of A"n": Aut(A"n") ≅ S"n". Conjugation by even elements are inner automorphisms of A"n" while the outer automorphism of A"n" of order 2 corresponds to conjugation by an odd element. For "n" = 6, there is an exceptional outer automorphism of A"n" so S"n" is not the full automorphism group of A"n".
Conversely, for "n" ≠ 6, S"n" has no outer automorphisms, and for "n" ≠ 2 it has no center, so for "n" ≠ 2, 6 it is a complete group, as discussed in automorphism group, below.
For "n" ≥ 5, S"n" is an almost simple group, as it lies between the simple group A"n" and its group of automorphisms.
S"n" can be embedded into A"n"+2 by appending the transposition ("n" + 1, "n" + 2) to all odd permutations, while embedding into A"n"+1 is impossible for "n" > 1.
Generators and relations.
The symmetric group on n letters is generated by the adjacent transpositions $ \sigma_i = (i, i + 1)$ that swap i and "i" + 1. The collection $\sigma_1, \ldots, \sigma_{n-1}$ generates S"n" subject to the following relations:
where 1 represents the identity permutation. This representation endows the symmetric group with the structure of a Coxeter group (and so also a reflection group).
Other possible generating sets include the set of transpositions that swap 1 and i for 2 ≤ "i" ≤ "n", and a set containing any n-cycle and a 2-cycle of adjacent elements in the n-cycle.
Subgroup structure.
A subgroup of a symmetric group is called a permutation group.
Normal subgroups.
The normal subgroups of the finite symmetric groups are well understood. If "n" ≤ 2, S"n" has at most 2 elements, and so has no nontrivial proper subgroups. The alternating group of degree "n" is always a normal subgroup, a proper one for "n" ≥ 2 and nontrivial for "n" ≥ 3; for "n" ≥ 3 it is in fact the only nontrivial proper normal subgroup of S"n", except when "n" = 4 where there is one additional such normal subgroup, which is isomorphic to the Klein four group.
The symmetric group on an infinite set does not have a subgroup of index 2, as Vitali (1915) proved that each permutation can be written as a product of three squares. However it contains the normal subgroup "S" of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of "S" that are products of an even number of transpositions form a subgroup of index 2 in "S", called the alternating subgroup "A". Since "A" is even a characteristic subgroup of "S", it is also a normal subgroup of the full symmetric group of the infinite set. The groups "A" and "S" are the only nontrivial proper normal subgroups of the symmetric group on a countably infinite set. This was first proved by Onofri (1929) and independently Schreier–Ulam (1934). For more details see or .
Maximal subgroups.
The maximal subgroups of S"n" fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the form S"k" × S"n"–"k" for 1 ≤ "k" < "n"/2. The imprimitive maximal subgroups are exactly those of the form S"k" wr S"n"/"k", where 2 ≤ "k" ≤ "n"/2 is a proper divisor of "n" and "wr" denotes the wreath product. The primitive maximal subgroups are more difficult to identify, but with the assistance of the O'Nan–Scott theorem and the classification of finite simple groups, gave a fairly satisfactory description of the maximal subgroups of this type, according to .
Sylow subgroups.
The Sylow subgroups of the symmetric groups are important examples of "p"-groups. They are more easily described in special cases first:
The Sylow "p"-subgroups of the symmetric group of degree "p" are just the cyclic subgroups generated by "p"-cycles. There are ("p" − 1)!/("p" − 1) = ("p" − 2)! such subgroups simply by counting generators. The normalizer therefore has order "p"⋅("p" − 1) and is known as a Frobenius group "F""p"("p"−1) (especially for "p" = 5), and is the affine general linear group, AGL(1, "p").
The Sylow "p"-subgroups of the symmetric group of degree "p"2 are the wreath product of two cyclic groups of order "p". For instance, when "p" = 3, a Sylow 3-subgroup of Sym(9) is generated by "a" = (1 4 7)(2 5 8)(3 6 9) and the elements "x" = (1 2 3), "y" = (4 5 6), "z" = (7 8 9), and every element of the Sylow 3-subgroup has the form "a""i""x""j""y""k""z""l" for .
The Sylow "p"-subgroups of the symmetric group of degree "p""n" are sometimes denoted W"p"("n"), and using this notation one has that W"p"("n" + 1) is the wreath product of W"p"("n") and W"p"(1).
In general, the Sylow "p"-subgroups of the symmetric group of degree "n" are a direct product of "a""i" copies of W"p"("i"), where 0 ≤ "ai" ≤ "p" − 1 and "n" = "a"0 + "p"⋅"a"1 + ... + "p""k"⋅"a""k" (the base "p" expansion of "n").
For instance, W2(1) = C2 and W2(2) = D8, the dihedral group of order 8, and so a Sylow 2-subgroup of the symmetric group of degree 7 is generated by { (1,3)(2,4), (1,2), (3,4), (5,6) } and is isomorphic to D8 × C2.
These calculations are attributed to and described in more detail in . Note however that attributes the result to an 1844 work of Cauchy, and mentions that it is even covered in textbook form in .
Transitive subgroups.
A transitive subgroup of S"n" is a subgroup whose action on {1, 2, ..., "n"} is transitive. For example, the Galois group of a (finite) Galois extension is a transitive subgroup of S"n", for some "n".
Cayley's theorem.
Cayley's theorem states that every group "G" is isomorphic to a subgroup of some symmetric group. In particular, one may take a subgroup of the symmetric group on the elements of "G", since every group acts on itself faithfully by (left or right) multiplication.
Cyclic subgroups.
Cyclic groups are those that are generated by a single permutation. When a permutation is represented in cycle notation, the order of the cyclic subgroup that it generates is the least common multiple of the lengths of its cycles. For example, in S5, one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S5 are generated by elements like (123)(45) that have one cycle of length 3 and another cycle of length 2. This rules out many groups as possible subgroups of symmetric groups of a given size. For example, S5 has no subgroup of order 15 (a divisor of the order of S5), because the only group of order 15 is the cyclic group. The largest possible order of a cyclic subgroup (equivalently, the largest possible order of an element in S"n") is given by Landau's function.
Automorphism group.
For "n" ≠ 2, 6, S"n" is a complete group: its center and outer automorphism group are both trivial.
For "n" = 2, the automorphism group is trivial, but S2 is not trivial: it is isomorphic to C2, which is abelian, and hence the center is the whole group.
For "n" = 6, it has an outer automorphism of order 2: Out(S6) = C2, and the automorphism group is a semidirect product Aut(S6) = S6 ⋊ C2.
In fact, for any set "X" of cardinality other than 6, every automorphism of the symmetric group on "X" is inner, a result first due to according to .
Homology.
The group homology of S"n" is quite regular and stabilizes: the first homology (concretely, the abelianization) is:
$H_1(\mathrm{S}_n,\mathbf{Z}) = \begin{cases} 0 & n < 2\\
\mathbf{Z}/2 & n \geq 2.\end{cases}$
The first homology group is the abelianization, and corresponds to the sign map S"n" → S2 which is the abelianization for "n" ≥ 2; for "n" < 2 the symmetric group is trivial. This homology is easily computed as follows: S"n" is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps S"n" → C"p" are to S2 and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps S"n" → S2 ≅ {±1} send an involution to 1 (the trivial map) or to −1 (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of S"n".
The second homology (concretely, the Schur multiplier) is:
$H_2(\mathrm{S}_n,\mathbf{Z}) = \begin{cases} 0 & n < 4\\
\mathbf{Z}/2 & n \geq 4.\end{cases}$
This was computed in , and corresponds to the double cover of the symmetric group, 2 · S"n".
Note that the exceptional low-dimensional homology of the alternating group ($H_1(\mathrm{A}_3)\cong H_1(\mathrm{A}_4) \cong \mathrm{C}_3,$ corresponding to non-trivial abelianization, and $H_2(\mathrm{A}_6)\cong H_2(\mathrm{A}_7) \cong \mathrm{C}_6,$ due to the exceptional 3-fold cover) does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map $\mathrm{A}_4 \twoheadrightarrow \mathrm{C}_3$ extends to $\mathrm{S}_4 \twoheadrightarrow \mathrm{S}_3,$ and the triple covers of A6 and A7 extend to triple covers of S6 and S7 – but these are not "homological" – the map $\mathrm{S}_4 \twoheadrightarrow \mathrm{S}_3$ does not change the abelianization of S4, and the triple covers do not correspond to homology either.
The homology "stabilizes" in the sense of stable homotopy theory: there is an inclusion map S"n" → S"n"+1, and for fixed "k", the induced map on homology "H""k"(S"n") → "H""k"(S"n"+1) is an isomorphism for sufficiently high "n". This is analogous to the homology of families Lie groups stabilizing.
The homology of the infinite symmetric group is computed in , with the cohomology algebra forming a Hopf algebra.
Representation theory.
The representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles.
The symmetric group S"n" has order "n"!. Its conjugacy classes are labeled by partitions of "n". Therefore, according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of "n". Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representation by the same set that parametrizes conjugacy classes, namely by partitions of "n" or equivalently Young diagrams of size "n".
Each such irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram.
Over other fields the situation can become much more complicated. If the field "K" has characteristic equal to zero or greater than "n" then by Maschke's theorem the group algebra "K"S"n" is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary).
However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of modules rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called "Specht modules", and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimensions are not known in general.
The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory.
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Cardinality of a mathematical group, or of the subgroup generated by an element
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is "infinite". The "order" of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element a of a group, is thus the smallest positive integer "m" such that "a""m" = "e", where "e" denotes the identity element of the group, and "a""m" denotes the product of "m" copies of "a". If no such "m" exists, the order of "a" is infinite.
The order of a group G is denoted by ord("G") or , and the order of an element "a" is denoted by ord("a") or , instead of $\operatorname{ord}(\langle a\rangle),$ where the brackets denote the generated group.
Lagrange's theorem states that for any subgroup "H" of a finite group "G", the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of .
Example.
The symmetric group S3 has the following multiplication table.
This group has six elements, so ord(S3) = 6. By definition, the order of the identity, "e", is one, since "e" 1 = "e". Each of "s", "t", and "w" squares to "e", so these group elements have order two: |"s"| = |"t"| = |"w"| = 2. Finally, "u" and "v" have order 3, since "u"3 = "vu" = "e", and "v"3 = "uv" = "e".
Order and structure.
The order of a group "G" and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the factorization of |"G"|, the more complicated the structure of "G".
For |"G"| = 1, the group is trivial. In any group, only the identity element "a = e" has ord("a)" = 1. If every non-identity element in "G" is equal to its inverse (so that "a"2 = "e"), then ord("a") = 2; this implies "G" is abelian since $ab=(ab)^{-1}=b^{-1}a^{-1}=ba$. The converse is not true; for example, the (additive) cyclic group Z6 of integers modulo 6 is abelian, but the number 2 has order 3:
$2+2+2=6 \equiv 0 \pmod {6}$.
The relationship between the two concepts of order is the following: if we write
$\langle a \rangle = \{ a^{k}\colon k \in \mathbb{Z} \} $
for the subgroup generated by "a", then
$\operatorname{ord} (a) = \operatorname{ord}(\langle a \rangle).$
For any integer "k", we have
"ak" = "e" if and only if ord("a") divides "k".
In general, the order of any subgroup of "G" divides the order of "G". More precisely: if "H" is a subgroup of "G", then
ord("G") / ord("H") = ["G" : "H"], where ["G" : "H"] is called the index of "H" in "G", an integer. This is Lagrange's theorem. (This is, however, only true when G has finite order. If ord("G") = ∞, the quotient ord("G") / ord("H") does not make sense.)
As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord(S3) = 6, the possible orders of the elements are 1, 2, 3 or 6.
The following partial converse is true for finite groups: if "d" divides the order of a group "G" and "d" is a prime number, then there exists an element of order "d" in "G" (this is sometimes called Cauchy's theorem). The statement does not hold for composite orders, e.g. the Klein four-group does not have an element of order four). This can be shown by inductive proof. The consequences of the theorem include: the order of a group "G" is a power of a prime "p" if and only if ord("a") is some power of "p" for every "a" in "G".
If "a" has infinite order, then all non-zero powers of "a" have infinite order as well. If "a" has finite order, we have the following formula for the order of the powers of "a":
ord("ak") = ord("a") / gcd(ord("a"), "k")
for every integer "k". In particular, "a" and its inverse "a"−1 have the same order.
In any group,
$ \operatorname{ord}(ab) = \operatorname{ord}(ba)$
There is no general formula relating the order of a product "ab" to the orders of "a" and "b". In fact, it is possible that both "a" and "b" have finite order while "ab" has infinite order, or that both "a" and "b" have infinite order while "ab" has finite order. An example of the former is "a"("x") = 2−"x", "b"("x") = 1−"x" with "ab"("x") = "x"−1 in the group $Sym(\mathbb{Z})$. An example of the latter is "a"("x") = "x"+1, "b"("x") = "x"−1 with "ab"("x") = "x". If "ab" = "ba", we can at least say that ord("ab") divides lcm(ord("a"), ord("b")). As a consequence, one can prove that in a finite abelian group, if "m" denotes the maximum of all the orders of the group's elements, then every element's order divides "m".
Counting by order of elements.
Suppose "G" is a finite group of order "n", and "d" is a divisor of "n". The number of order "d" elements in "G" is a multiple of φ("d") (possibly zero), where φ is Euler's totient function, giving the number of positive integers no larger than "d" and coprime to it. For example, in the case of S3, φ(3) = 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2) = 1, and is only of limited utility for composite "d" such as "d" = 6, since φ(6) = 2, and there are zero elements of order 6 in S3.
In relation to homomorphisms.
Group homomorphisms tend to reduce the orders of elements: if "f": "G" → "H" is a homomorphism, and "a" is an element of "G" of finite order, then ord("f"("a")) divides ord("a"). If "f" is injective, then ord("f"("a")) = ord("a"). This can often be used to prove that there are no homomorphisms or no injective homomorphisms, between two explicitly given groups. (For example, there can be no nontrivial homomorphism "h": S3 → Z5, because every number except zero in Z5 has order 5, which does not divide the orders 1, 2, and 3 of elements in S3.) A further consequence is that conjugate elements have the same order.
Class equation.
An important result about orders is the class equation; it relates the order of a finite group "G" to the order of its center Z("G") and the sizes of its non-trivial conjugacy classes:
$|G| = |Z(G)| + \sum_{i}d_i\;$
where the "di" are the sizes of the non-trivial conjugacy classes; these are proper divisors of |"G"| bigger than one, and they are also equal to the indices of the centralizers in "G" of the representatives of the non-trivial conjugacy classes. For example, the center of S3 is just the trivial group with the single element "e", and the equation reads |S3| = 1+2+3.
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In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements.
Formally, $\mathrm{S}_n$ is complete and the natural map $\mathrm{S}_n \to \operatorname{Aut}(\mathrm{S}_n)$ is an isomorphism.
Indeed, the natural maps $\mathrm{S}_n \to \operatorname{Aut}(\mathrm{S}_n) \to \operatorname{Aut}(\mathrm{A}_n)$ are isomorphisms.
$\operatorname{Aut}(\mathrm{S}_1)=\operatorname{Out}(\mathrm{S}_1)=\operatorname{Aut}(\mathrm{A}_1)=\operatorname{Out}(\mathrm{A}_1)=\mathrm{C}_1$
$\operatorname{Aut}(\mathrm{S}_2)=\operatorname{Out}(\mathrm{S}_2)=\operatorname{Aut}(\mathrm{A}_2)=\operatorname{Out}(\mathrm{A}_2)=\mathrm{C}_1$
The exceptional outer automorphism of S6.
Among symmetric groups, only S6 has a non-trivial outer automorphism,
which one can call "exceptional" (in analogy with exceptional Lie algebras) or "exotic". In fact, Out(S6) = C2.
This was discovered by Otto Hölder in 1895.
The specific nature of the outer automorphism is as follows:
Thus, all 720 permutations on 6 elements are accounted for. The outer automorphism does not preserve cycle structure in general, mapping single cycles to the product of two cycles and vice versa.
This also yields another outer automorphism of A6, and this is the only exceptional outer automorphism of a finite simple group: for the infinite families of simple groups, there are formulas for the number of outer automorphisms, and the simple group of order 360, thought of as A6, would be expected to have two outer automorphisms, not four.
However, when A6 is viewed as PSL(2, 9) the outer automorphism group has the expected order. (For sporadic groups – i.e. those not falling in an infinite family – the notion of exceptional outer automorphism is ill-defined, as there is no general formula.)
Construction.
There are numerous constructions, listed in .
Note that as an outer automorphism, it is a "class" of automorphisms, well-determined only up to an inner automorphism, hence there is not a natural one to write down.
One method is:
Throughout the following, one can work with the multiplication action on cosets or the conjugation action on conjugates.
To see that S6 has an outer automorphism, recall that homomorphisms
from a group "G" to a symmetric group S"n" are essentially the same as actions
of "G" on a set of "n" elements, and the subgroup fixing a point is then a subgroup of index at most "n" in "G". Conversely if we have a subgroup of index "n" in "G", the action on the cosets gives a transitive action of "G" on "n" points, and therefore a homomorphism to S"n".
Construction from graph partitions.
Before the more mathematically rigorous constructions, it helps to understand a simple construction.
Take a complete graph with 6 vertices, K6. It has 15 edges, which can be partitioned into perfect matchings in 15 different ways, each perfect matching being a set of three edges no two of which share a vertex. It is possible to find a set of 5 perfect matchings from the set of 15 such that no two matchings share an edge, and that between them include all 5 × 3 = 15 edges of the graph; this graph factorization can be done in 6 different ways.
Consider a permutation of the 6 vertices, and see its effect on the 6 different factorizations. We get a map from 720 input permutations to 720 output permutations. That map is precisely the outer automorphism of S6.
Being an automorphism, the map must preserve the order of elements, but unlike inner automorphisms, it does not preserve cycle structure, thereby indicating that it must be an outer automorphism. For instance, a 2-cycle maps to a product of three 2-cycles; it is easy to see that a 2-cycle affects all of the 6 graph factorizations in some way, and hence has no fixed points when viewed as a permutation of factorizations. The fact that it is possible to construct this automorphism at all relies on a large number of numerical coincidences which apply only to "n" = 6.
Exotic map S5 → S6.
There is a subgroup (indeed, 6 conjugate subgroups) of S6 which is abstractly isomorphic to S5, but which acts transitively as subgroups of S6 on a set of 6 elements. (The image of the obvious map S"n" → S"n"+1 fixes an element and thus is not transitive.)
Sylow 5-subgroups.
Janusz and Rotman construct it thus:
This follows from inspection of 5-cycles: each 5-cycle generates a group of order 5 (thus a Sylow subgroup), there are 5!/5 = 120/5 = 24 5-cycles, yielding 6 subgroups (as each subgroup also includes the identity), and S"n" acts transitively by conjugation on the set of cycles of a given class, hence transitively by conjugation on these subgroups.
Alternately, one could use the Sylow theorems, which state generally that all Sylow p-subgroups are conjugate.
PGL(2,5).
The projective linear group of dimension two over the finite field with five elements, PGL(2, 5), acts on the projective line over the field with five elements, P1(F5), which has six elements. Further, this action is faithful and 3-transitive, as is always the case for the action of the projective linear group on the projective line. This yields a map PGL(2, 5) → S6 as a transitive subgroup. Identifying PGL(2, 5) with S5 and the projective special linear group PSL(2, 5) with A5 yields the desired exotic maps S5 → S6 and A5 → A6.
Following the same philosophy, one can realize the outer automorphism as the following two inequivalent actions of S6 on a set with six elements:
Frobenius group.
Another way:
To construct an outer automorphism of S6, we need to construct
an "unusual" subgroup of index 6 in S6, in other words one that is not one of the six obvious S5 subgroups fixing a point (which just correspond to inner automorphisms of S6).
The Frobenius group of affine transformations of F5 (maps $x \mapsto ax+b$ where "a" ≠ 0) has order 20 = (5 − 1) · 5 and acts on the field with 5 elements, hence is a subgroup of S5.
S5 acts transitively on the coset space, which is a set of 120/20 = 6 elements (or by conjugation, which yields the action above).
Other constructions.
Ernst Witt found a copy of Aut(S6) in the Mathieu group M12 (a subgroup "T" isomorphic to S6 and an element "σ" that normalizes "T" and acts by outer automorphism). Similarly to S6 acting on a set of 6 elements in 2 different ways (having an outer automorphism), M12 acts on a set of 12 elements in 2 different ways (has an outer automorphism), though since "M"12 is itself exceptional, one does not consider this outer automorphism to be exceptional itself.
The full automorphism group of A6 appears naturally as a maximal subgroup of the Mathieu group M12 in 2 ways, as either a subgroup fixing a division of the 12 points into a pair of 6-element sets, or as a subgroup fixing a subset of 2 points.
Another way to see that S6 has a nontrivial outer automorphism is to use the fact that A6 is isomorphic to PSL2(9), whose automorphism group is the projective semilinear group PΓL2(9), in which PSL2(9) is of index 4, yielding an outer automorphism group of order 4. The most visual way to see this automorphism is to give an interpretation via algebraic geometry over finite fields, as follows. Consider the action of S6 on affine 6-space over the field k with 3 elements. This action preserves several things: the hyperplane "H" on which the coordinates sum to 0, the line "L" in "H" where all coordinates coincide, and the quadratic form "q" given by the sum of the squares of all 6 coordinates. The restriction of "q" to "H" has defect line "L", so there is an induced quadratic form "Q" on the 4-dimensional "H"/"L" that one checks is non-degenerate and non-split. The zero scheme of "Q" in "H"/"L" defines a smooth quadric surface "X" in the associated projective 3-space over "k". Over an algebraic closure of "k", "X" is a product of two projective lines, so by a descent argument "X" is the Weil restriction to "k" of the projective line over a quadratic étale algebra "K". Since "Q" is not split over "k", an auxiliary argument with special orthogonal groups over "k" forces "K" to be a field (rather than a product of two copies of "k"). The natural S6-action on everything in sight defines a map from S6 to the "k"-automorphism group of "X", which is the semi-direct product "G" of PGL2("K") = PGL2(9) against the Galois involution. This map carries the simple group A6 nontrivially into (hence onto) the subgroup PSL2(9) of index 4 in the semi-direct product "G", so S6 is thereby identified as an index-2 subgroup of "G" (namely, the subgroup of "G" generated by PSL2(9) and the Galois involution). Conjugation by any element of "G" outside of S6 defines the nontrivial outer automorphism of S6.
Structure of outer automorphism.
On cycles, it exchanges permutations of type (12) with (12)(34)(56) (class 21 with class 23), and of type (123) with (145)(263) (class 31 with class 32). The outer automorphism also exchanges permutations of type (12)(345) with (123456) (class 2131 with class 61). For each of the other cycle types in S6, the outer automorphism fixes the class of permutations of the cycle type.
On A6, it interchanges the 3-cycles (like (123)) with elements of class 32 (like (123)(456)).
No other outer automorphisms.
To see that none of the other symmetric groups have outer automorphisms, it is easiest to proceed in two steps:
The latter can be shown in two ways:
Each permutation of order two (called an involution) is a product of "k" > 0 disjoint transpositions, so that it has cyclic structure 2"k"1"n"−2"k". What is special about the class of transpositions ("k" = 1)?
If one forms the product of two distinct transpositions "τ"1 and "τ"2, then one always obtains either a 3-cycle or a permutation of type 221"n"−4, so the order of the produced element is either 2 or 3. On the other hand, if one forms the product of two distinct involutions "σ"1, "σ"2 of type "k" > 1, then provided "n" ≥ 7, it is always possible to produce an element of order 6, 7 or 4, as follows. We can arrange that the product contains either
For "k" ≥ 5, adjoin to the permutations "σ"1, "σ"2 of the last example redundant 2-cycles that cancel each other, and we still get two 4-cycles.
Now we arrive at a contradiction, because if the class of transpositions is sent via the automorphism "f" to a class of involutions that has "k" > 1, then there exist two transpositions "τ"1, "τ"2 such that "f"("τ"1) "f"("τ"2) has order 6, 7 or 4, but we know that "τ"1"τ"2 has order 2 or 3.
No other outer automorphisms of S6.
S6 has exactly one (class) of outer automorphisms: Out(S6) = C2.
To see this, observe that there are only two conjugacy classes of S6 of size 15: the transpositions and those of class 23. Each element of Aut(S6) either preserves each of these conjugacy classes, or exchanges them. Any representative of the outer automorphism constructed above exchanges the conjugacy classes, whereas an index 2 subgroup stabilizes the transpositions. But an automorphism that stabilizes the transpositions is inner, so the inner automorphisms form an index 2 subgroup of Aut(S6), so Out(S6) = C2.
More pithily: an automorphism that stabilizes transpositions is inner, and there are only two conjugacy classes of order 15 (transpositions and triple transpositions), hence the outer automorphism group is at most order 2.
Small "n".
Symmetric.
For "n" = 2, S2 = C2 = Z/2 and the automorphism group is trivial (obviously, but more formally because Aut(Z/2) = GL(1, Z/2) = Z/2* = C1). The inner automorphism group is thus also trivial (also because S2 is abelian).
Alternating.
For "n" = 1 and 2, A1 = A2 = C1 is trivial, so the automorphism group is also trivial. For "n" = 3, A3 = C3 = Z/3 is abelian (and cyclic): the automorphism group is GL(1, Z/3*) = C2, and the inner automorphism group is trivial (because it is abelian).
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Group in which the order of every element is a power of p
In mathematics, specifically group theory, given a prime number "p", a "p"-group is a group in which the order of every element is a power of "p". That is, for each element "g" of a "p"-group "G", there exists a nonnegative integer "n" such that the product of "pn" copies of "g", and not fewer, is equal to the identity element. The orders of different elements may be different powers of "p".
Abelian "p"-groups are also called "p"-primary or simply primary.
A finite group is a "p"-group if and only if its order (the number of its elements) is a power of "p". Given a finite group "G", the Sylow theorems guarantee the existence of a subgroup of "G" of order "pn" for every prime power "pn" that divides the order of "G".
Every finite "p"-group is nilpotent.
The remainder of this article deals with finite "p"-groups. For an example of an infinite abelian "p"-group, see Prüfer group, and for an example of an infinite simple "p"-group, see Tarski monster group.
Properties.
Every "p"-group is periodic since by definition every element has finite order.
If "p" is prime and "G" is a group of order "p""k", then "G" has a normal subgroup of order "p""m" for every 1 ≤ "m" ≤ "k". This follows by induction, using Cauchy's theorem and the Correspondence Theorem for groups. A proof sketch is as follows: because the center "Z" of "G" is non-trivial (see below), according to Cauchy's theorem "Z" has a subgroup "H" of order "p". Being central in "G", "H" is necessarily normal in "G". We may now apply the inductive hypothesis to "G/H", and the result follows from the Correspondence Theorem.
Non-trivial center.
One of the first standard results using the class equation is that the center of a non-trivial finite "p"-group cannot be the trivial subgroup.
This forms the basis for many inductive methods in "p"-groups.
For instance, the normalizer "N" of a proper subgroup "H" of a finite "p"-group "G" properly contains "H", because for any counterexample with "H" = "N", the center "Z" is contained in "N", and so also in "H", but then there is a smaller example "H"/"Z" whose normalizer in "G"/"Z" is "N"/"Z" = "H"/"Z", creating an infinite descent. As a corollary, every finite "p"-group is nilpotent.
In another direction, every normal subgroup "N" of a finite "p"-group intersects the center non-trivially as may be proved by considering the elements of "N" which are fixed when "G" acts on "N" by conjugation. Since every central subgroup is normal, it follows that every minimal normal subgroup of a finite "p"-group is central and has order "p". Indeed, the socle of a finite "p"-group is the subgroup of the center consisting of the central elements of order "p".
If "G" is a "p"-group, then so is "G"/"Z", and so it too has a non-trivial center. The preimage in "G" of the center of "G"/"Z" is called the second center and these groups begin the upper central series. Generalizing the earlier comments about the socle, a finite "p"-group with order "pn" contains normal subgroups of order "pi" with 0 ≤ "i" ≤ "n", and any normal subgroup of order "pi" is contained in the "i"th center "Z""i". If a normal subgroup is not contained in "Z""i", then its intersection with "Z""i"+1 has size at least "p""i"+1.
Automorphisms.
The automorphism groups of "p"-groups are well studied. Just as every finite "p"-group has a non-trivial center so that the inner automorphism group is a proper quotient of the group, every finite "p"-group has a non-trivial outer automorphism group. Every automorphism of "G" induces an automorphism on "G"/Φ("G"), where Φ("G") is the Frattini subgroup of "G". The quotient G/Φ("G") is an elementary abelian group and its automorphism group is a general linear group, so very well understood. The map from the automorphism group of "G" into this general linear group has been studied by Burnside, who showed that the kernel of this map is a "p"-group.
Examples.
"p"-groups of the same order are not necessarily isomorphic; for example, the cyclic group "C"4 and the Klein four-group "V"4 are both 2-groups of order 4, but they are not isomorphic.
Nor need a "p"-group be abelian; the dihedral group Dih4 of order 8 is a non-abelian 2-group. However, every group of order "p"2 is abelian.
The dihedral groups are both very similar to and very dissimilar from the quaternion groups and the semidihedral groups. Together the dihedral, semidihedral, and quaternion groups form the 2-groups of maximal class, that is those groups of order 2"n"+1 and nilpotency class "n".
Iterated wreath products.
The iterated wreath products of cyclic groups of order "p" are very important examples of "p"-groups. Denote the cyclic group of order "p" as "W"(1), and the wreath product of "W"("n") with "W"(1) as "W"("n" + 1). Then "W"("n") is the Sylow "p"-subgroup of the symmetric group Sym("p""n"). Maximal "p"-subgroups of the general linear group GL("n",Q) are direct products of various "W"("n"). It has order "p""k" where "k" = ("p""n" − 1)/("p" − 1). It has nilpotency class "p""n"−1, and its lower central series, upper central series, lower exponent-"p" central series, and upper exponent-"p" central series are equal. It is generated by its elements of order "p", but its exponent is "p""n". The second such group, "W"(2), is also a "p"-group of maximal class, since it has order "p""p"+1 and nilpotency class "p", but is not a regular "p"-group. Since groups of order "p""p" are always regular groups, it is also a minimal such example.
Generalized dihedral groups.
When "p" = 2 and "n" = 2, "W"("n") is the dihedral group of order 8, so in some sense "W"("n") provides an analogue for the dihedral group for all primes "p" when "n" = 2. However, for higher "n" the analogy becomes strained. There is a different family of examples that more closely mimics the dihedral groups of order 2"n", but that requires a bit more setup. Let ζ denote a primitive "p"th root of unity in the complex numbers, let Z[ζ] be the ring of cyclotomic integers generated by it, and let "P" be the prime ideal generated by 1−ζ. Let "G" be a cyclic group of order "p" generated by an element "z". Form the semidirect product "E"("p") of Z[ζ] and "G" where "z" acts as multiplication by ζ. The powers "P""n" are normal subgroups of "E"("p"), and the example groups are "E"("p","n") = "E"("p")/"P""n". "E"("p","n") has order "p""n"+1 and nilpotency class "n", so is a "p"-group of maximal class. When "p" = 2, "E"(2,"n") is the dihedral group of order 2"n". When "p" is odd, both "W"(2) and "E"("p","p") are irregular groups of maximal class and order "p""p"+1, but are not isomorphic.
Unitriangular matrix groups.
The Sylow subgroups of general linear groups are another fundamental family of examples. Let "V" be a vector space of dimension "n" with basis { "e"1, "e"2, ..., "e""n" } and define "V""i" to be the vector space generated by { "e""i", "e""i"+1, ..., "e""n" } for 1 ≤ "i" ≤ "n", and define "V""i" = 0 when "i" > "n". For each 1 ≤ "m" ≤ "n", the set of invertible linear transformations of "V" which take each "V""i" to "V""i"+"m" form a subgroup of Aut("V") denoted "U""m". If "V" is a vector space over Z/"p"Z, then "U"1 is a Sylow "p"-subgroup of Aut("V") = GL("n", "p"), and the terms of its lower central series are just the "U""m". In terms of matrices, "U""m" are those upper triangular matrices with 1s one the diagonal and 0s on the first "m"−1 superdiagonals. The group "U"1 has order "p""n"·("n"−1)/2, nilpotency class "n", and exponent "p""k" where "k" is the least integer at least as large as the base "p" logarithm of "n".
Classification.
The groups of order "p""n" for 0 ≤ "n" ≤ 4 were classified early in the history of group theory, and modern work has extended these classifications to groups whose order divides "p"7, though the sheer number of families of such groups grows so quickly that further classifications along these lines are judged difficult for the human mind to comprehend. For example, Marshall Hall Jr. and James K. Senior classified groups of order 2"n" for "n" ≤ 6 in 1964.
Rather than classify the groups by order, Philip Hall proposed using a notion of isoclinism of groups which gathered finite "p"-groups into families based on large quotient and subgroups.
An entirely different method classifies finite "p"-groups by their coclass, that is, the difference between their composition length and their nilpotency class. The so-called coclass conjectures described the set of all finite "p"-groups of fixed coclass as perturbations of finitely many pro-p groups. The coclass conjectures were proven in the 1980s using techniques related to Lie algebras and powerful p-groups. The final proofs of the coclass theorems are due to A. Shalev and independently to C. R. Leedham-Green, both in 1994. They admit a classification of finite "p"-groups in directed coclass graphs consisting of only finitely many coclass trees whose (infinitely many) members are characterized by finitely many parametrized presentations.
Every group of order "p"5 is metabelian.
Up to "p"3.
The trivial group is the only group of order one, and the cyclic group C"p" is the only group of order "p". There are exactly two groups of order "p"2, both abelian, namely C"p"2 and C"p" × C"p". For example, the cyclic group C4 and the Klein four-group "V"4 which is C2 × C2 are both 2-groups of order 4.
There are three abelian groups of order "p"3, namely C"p"3, C"p"2 × C"p", and C"p" × C"p" × C"p". There are also two non-abelian groups.
For "p" ≠ 2, one is a semi-direct product of C"p" × C"p" with C"p", and the other is a semi-direct product of C"p"2 with C"p". The first one can be described in other terms as group UT(3,"p") of unitriangular matrices over finite field with "p" elements, also called the Heisenberg group mod "p".
For "p" = 2, both the semi-direct products mentioned above are isomorphic to the dihedral group Dih4 of order 8. The other non-abelian group of order 8 is the quaternion group Q8.
Prevalence.
Among groups.
The number of isomorphism classes of groups of order "pn" grows as $p^{\frac{2}{27}n^3+O(n^{8/3})}$, and these are dominated by the classes that are two-step nilpotent. Because of this rapid growth, there is a folklore conjecture asserting that almost all finite groups are 2-groups: the fraction of isomorphism classes of 2-groups among isomorphism classes of groups of order at most "n" is thought to tend to 1 as "n" tends to infinity. For instance, of the 49 910 529 484 different groups of order at most 2000, , or just over 99%, are 2-groups of order 1024.
Within a group.
Every finite group whose order is divisible by "p" contains a subgroup which is a non-trivial "p"-group, namely a cyclic group of order "p" generated by an element of order "p" obtained from Cauchy's theorem. In fact, it contains a "p"-group of maximal possible order: if $|G|=n=p^km$ where "p" does not divide "m," then "G" has a subgroup "P" of order $p^k,$ called a Sylow "p"-subgroup. This subgroup need not be unique, but any subgroups of this order are conjugate, and any "p"-subgroup of "G" is contained in a Sylow "p"-subgroup. This and other properties are proved in the Sylow theorems.
Application to structure of a group.
"p"-groups are fundamental tools in understanding the structure of groups and in the classification of finite simple groups. "p"-groups arise both as subgroups and as quotient groups. As subgroups, for a given prime "p" one has the Sylow "p"-subgroups "P" (largest "p"-subgroup not unique but all conjugate) and the "p"-core $O_p(G)$ (the unique largest "normal" "p"-subgroup), and various others. As quotients, the largest "p"-group quotient is the quotient of "G" by the "p"-residual subgroup $O^p(G).$ These groups are related (for different primes), possess important properties such as the focal subgroup theorem, and allow one to determine many aspects of the structure of the group.
Local control.
Much of the structure of a finite group is carried in the structure of its so-called local subgroups, the normalizers of non-identity "p"-subgroups.
The large elementary abelian subgroups of a finite group exert control over the group that was used in the proof of the Feit–Thompson theorem. Certain central extensions of elementary abelian groups called extraspecial groups help describe the structure of groups as acting on symplectic vector spaces.
Richard Brauer classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, and John Walter, Daniel Gorenstein, Helmut Bender, Michio Suzuki, George Glauberman, and others classified those simple groups whose Sylow 2-subgroups were abelian, dihedral, semidihedral, or quaternion.
Footnotes.
Notes.
Citations.
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Mathematical group that can be generated as the set of powers of a single element
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C"n", that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element "g" such that every other element of the group may be obtained by repeatedly applying the group operation to "g" or its inverse. Each element can be written as an integer power of "g" in multiplicative notation, or as an integer multiple of "g" in additive notation. This element "g" is called a "generator" of the group.
Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order "n" is isomorphic to the additive group of Z/"n"Z, the integers modulo "n". Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.
Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.
Definition and notation.
For any element "g" in any group "G", one can form the subgroup that consists of all its integer powers: ⟨"g"⟩ = {"g""k" | "k" ∈ Z}, called the cyclic subgroup generated by "g". The order of "g" is |⟨"g"⟩|, the number of elements in ⟨"g"⟩, conventionally abbreviated as |"g"|, as ord("g"), or as o("g"). That is, the order of an element is equal to the order of the cyclic subgroup that it generates,
A "cyclic group" is a group which is equal to one of its cyclic subgroups: "G" = ⟨"g"⟩ for some element "g", called a "generator" of "G".
For a finite cyclic group "G" of order "n" we have "G" = {"e", "g", "g"2, ... , "g""n"−1}, where "e" is the identity element and "g""i" = "g""j" whenever "i" ≡ "j" (mod "n"); in particular "g""n" = "g"0 = "e", and "g"−1 = "g""n"−1. An abstract group defined by this multiplication is often denoted C"n", and we say that "G" is isomorphic to the standard cyclic group C"n". Such a group is also isomorphic to Z/"n"Z, the group of integers modulo "n" with the addition operation, which is the standard cyclic group in additive notation. Under the isomorphism "χ" defined by "χ"("g""i") = "i" the identity element "e" corresponds to 0, products correspond to sums, and powers correspond to multiples.
For example, the set of complex 6th roots of unity <math display="block">G = \left\{\pm 1, \pm{ \left(\tfrac 1 2 + \tfrac{\sqrt 3}{2}i\right)}, \pm{\left(\tfrac 1 2 - \tfrac{\sqrt 3}{2}i\right)}\right\}$ forms a group under multiplication. It is cyclic, since it is generated by the primitive root $z = \tfrac 1 2 + \tfrac{\sqrt 3}{2}i=e^{2\pi i/6}:$ that is, "G" = ⟨"z"⟩ = { 1, "z", "z"2, "z"3, "z"4, "z"5 } with "z"6 = 1. Under a change of letters, this is isomorphic to (structurally the same as) the standard cyclic group of order 6, defined as C6 = ⟨"g"⟩ = {"e", "g", "g"2, "g"3, "g"4, "g"5} with multiplication "g""j" · "g""k" = "g""j"+"k" (mod 6), so that "g"6 = "g"0 = "e". These groups are also isomorphic to Z/6Z = {0, 1, 2, 3, 4, 5} with the operation of addition modulo 6, with "z""k" and "g""k" corresponding to "k". For example, 1 + 2 ≡ 3 (mod 6) corresponds to "z"1 · "z"2 = "z"3, and 2 + 5 ≡ 1 (mod 6) corresponds to "z"2 · "z"5 = "z"7 = "z"1, and so on. Any element generates its own cyclic subgroup, such as ⟨"z"2⟩ = {"e", "z"2, "z"4} of order 3, isomorphic to C3 and Z/3Z; and ⟨"z"5⟩ = { "e", "z"5, "z"10 = "z"4, "z"15 = "z"3, "z"20 = "z"2, "z"25 = "z" } = "G", so that "z"5 has order 6 and is an alternative generator of "G".
Instead of the quotient notations Z/"n"Z, Z/("n"), or Z/"n", some authors denote a finite cyclic group as Z"n", but this clashes with the notation of number theory, where Z"p" denotes a "p"-adic number ring, or localization at a prime ideal.
On the other hand, in an infinite cyclic group "G" = ⟨"g"⟩, the powers "g""k" give distinct elements for all integers "k", so that "G" = {... , "g"−2, "g"−1, "e", "g", "g"2, ...}, and "G" is isomorphic to the standard group C = C∞ and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, and the name "cyclic" may be misleading.
To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group".
Examples.
Integer and modular addition.
The set of integers Z, with the operation of addition, forms a group. It is an infinite cyclic group, because all integers can be written by repeatedly adding or subtracting the single number 1. In this group, 1 and −1 are the only generators. Every infinite cyclic group is isomorphic to Z.
For every positive integer "n", the set of integers modulo "n", again with the operation of addition, forms a finite cyclic group, denoted Z/"n"Z.
A modular integer "i" is a generator of this group if "i" is relatively prime to "n", because these elements can generate all other elements of the group through integer addition.
Every finite cyclic group "G" is isomorphic to Z/"n"Z, where "n" = |"G"| is the order of the group.
The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of commutative rings, also denoted Z and Z/"nZ or Z/("n"). If "p" is a prime, then Z/"pZ is a finite field, and is usually denoted F"p" or GF("p") for Galois field.
Modular multiplication.
For every positive integer "n", the set of the integers modulo "n" that are relatively prime to "n" is written as (Z/"n"Z)×; it forms a group under the operation of multiplication. This group is not always cyclic, but is so whenever "n" is 1, 2, 4, a power of an odd prime, or twice a power of an odd prime (sequence in the OEIS).
This is the multiplicative group of units of the ring Z/"nZ; there are "φ"("n") of them, where again "φ" is the Euler totient function. For example, (Z/6Z)× = {1, 5}, and since 6 is twice an odd prime this is a cyclic group. In contrast, (Z/8Z)× = {1, 3, 5, 7} is a Klein 4-group and is not cyclic. When (Z/"nZ)× is cyclic, its generators are called primitive roots modulo "n".
For a prime number "p", the group (Z/"p"Z)× is always cyclic, consisting of the non-zero elements of the finite field of order "p". More generally, every finite subgroup of the multiplicative group of any field is cyclic.
Rotational symmetries.
The set of rotational symmetries of a polygon forms a finite cyclic group. If there are "n" different ways of moving the polygon to itself by a rotation (including the null rotation) then this symmetry group is isomorphic to Z/"n"Z. In three or higher dimensions there exist other finite symmetry groups that are cyclic, but which are not all rotations around an axis, but instead rotoreflections.
The group of all rotations of a circle (the circle group, also denoted "S"1) is "not" cyclic, because there is no single rotation whose integer powers generate all rotations. In fact, the infinite cyclic group C∞ is countable, while "S"1 is not. The group of rotations by rational angles "is" countable, but still not cyclic.
Galois theory.
An "n"th root of unity is a complex number whose "n"th power is 1, a root of the polynomial "x""n" − 1. The set of all "n"th roots of unity forms a cyclic group of order "n" under multiplication. The generators of this cyclic group are the "n"th primitive roots of unity; they are the roots of the "n"th cyclotomic polynomial.
For example, the polynomial "z"3 − 1 factors as ("z" − 1)("z" − "ω")("z" − "ω"2), where "ω" = "e"2"πi"/3; the set {1, "ω", "ω"2} = {"ω"0, "ω"1, "ω"2} forms a cyclic group under multiplication. The Galois group of the field extension of the rational numbers generated by the "n"th roots of unity forms a different group, isomorphic to the multiplicative group (Z/nZ)× of order "φ"("n"), which is cyclic for some but not all "n" (see above).
A field extension is called a cyclic extension if its Galois group is cyclic. For fields of characteristic zero, such extensions are the subject of Kummer theory, and are intimately related to solvability by radicals. For an extension of finite fields of characteristic "p", its Galois group is always finite and cyclic, generated by a power of the Frobenius mapping. Conversely, given a finite field "F" and a finite cyclic group "G", there is a finite field extension of "F" whose Galois group is "G".
Subgroups.
All subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form ⟨"m"⟩ = "mZ, with "m" a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0Z, they all are isomorphic to Z. The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility. Thus, since a prime number "p" has no nontrivial divisors, "pZ is a maximal proper subgroup, and the quotient group Z/"p"Z is simple; in fact, a cyclic group is simple if and only if its order is prime.
All quotient groups Z/"nZ are finite, with the exception Z/0Z = Z/{0}. For every positive divisor "d" of "n", the quotient group Z/"nZ has precisely one subgroup of order "d", generated by the residue class of "n"/"d". There are no other subgroups.
Additional properties.
Every cyclic group is abelian. That is, its group operation is commutative: "gh" = "hg" (for all "g" and "h" in "G"). This is clear for the groups of integer and modular addition since "r" + "s" ≡ "s" + "r" (mod "n"), and it follows for all cyclic groups since they are all isomorphic to these standard groups. For a finite cyclic group of order "n", "g""n" is the identity element for any element "g". This again follows by using the isomorphism to modular addition, since "kn" ≡ 0 (mod "n") for every integer "k". (This is also true for a general group of order "n", due to Lagrange's theorem.)
For a prime power $p^k$, the group $Z/p^k Z$ is called a primary cyclic group. The fundamental theorem of abelian groups states that every finitely generated abelian group is a finite direct product of primary cyclic and infinite cyclic groups.
Because a cyclic group $G $ is abelian, the conjugate class $\text{Cl}(x)=\{gxg^{-1}: g\in G\}=\{x\} $ for $x \in G $. Thus, each of its conjugacy classes consists of a single element. A cyclic group of order "n" therefore has "n" conjugacy classes.
If "d" is a divisor of "n", then the number of elements in Z/"n"Z which have order "d" is "φ"("d"), and the number of elements whose order divides "d" is exactly "d".
If "G" is a finite group in which, for each "n" > 0, "G" contains at most "n" elements of order dividing "n", then "G" must be cyclic.
The order of an element "m" in Z/"n"Z is "n"/gcd("n","m").
If "n" and "m" are coprime, then the direct product of two cyclic groups Z/"nZ and Z/"mZ is isomorphic to the cyclic group Z/"nm"Z, and the converse also holds: this is one form of the Chinese remainder theorem. For example, Z/12Z is isomorphic to the direct product Z/3Z × Z/4Z under the isomorphism ("k" mod 12) → ("k" mod 3, "k" mod 4); but it is not isomorphic to Z/6Z × Z/2Z, in which every element has order at most 6.
If "p" is a prime number, then any group with "p" elements is isomorphic to the simple group Z/"p"Z.
A number "n" is called a cyclic number if Z/"n"Z is the only group of order "n", which is true exactly when gcd("n", "φ"("n")) = 1. The sequence of cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd except 2. The cyclic numbers are:
1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ... (sequence in the OEIS)
The definition immediately implies that cyclic groups have group presentation C∞ = ⟨"x" | ⟩ and C"n" = ⟨"x" | "x""n"⟩ for finite "n".
Associated objects.
Representations.
The representation theory of the cyclic group is a critical base case for the representation theory of more general finite groups. In the complex case, a representation of a cyclic group decomposes into a direct sum of linear characters, making the connection between character theory and representation theory transparent. In the positive characteristic case, the indecomposable representations of the cyclic group form a model and inductive basis for the representation theory of groups with cyclic Sylow subgroups and more generally the representation theory of blocks of cyclic defect.
Cycle graph.
A cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle graph for a cyclic group is simply a circular graph, where the group order is equal to the number of nodes. A single generator defines the group as a directional path on the graph, and the inverse generator defines a backwards path. A trivial path (identity) can be drawn as a loop but is usually suppressed. Z2 is sometimes drawn with two curved edges as a multigraph.
A cyclic group Z"n", with order "n", corresponds to a single cycle graphed simply as an "n"-sided polygon with the elements at the vertices.
Cayley graph.
A Cayley graph is a graph defined from a pair ("G","S") where "G" is a group and "S" is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a cycle graph, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite path graph. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called circulant graphs. These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point. They are exactly the vertex-transitive graphs whose symmetry group includes a transitive cyclic group.
Endomorphisms.
The endomorphism ring of the abelian group Z/"nZ is isomorphic to Z/"nZ itself as a ring. Under this isomorphism, the number "r" corresponds to the endomorphism of Z/"nZ that maps each element to the sum of "r" copies of it. This is a bijection if and only if "r" is coprime with "n", so the automorphism group of Z/"nZ is isomorphic to the unit group (Z/"n"Z)×.
Similarly, the endomorphism ring of the additive group of Z is isomorphic to the ring Z. Its automorphism group is isomorphic to the group of units of the ring Z, which is ({−1, +1}, ×) ≅ C2.
Related classes of groups.
Several other classes of groups have been defined by their relation to the cyclic groups:
Virtually cyclic groups.
A group is called virtually cyclic if it contains a cyclic subgroup of finite index (the number of cosets that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by multiplying a member of the cyclic subgroup and a member of a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; an example of such a group is the direct product of Z/"n"Z and Z, in which the factor Z has finite index "n". Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.
Locally cyclic groups.
A locally cyclic group is a group in which each finitely generated subgroup is cyclic. An example is the additive group of the rational numbers: every finite set of rational numbers is a set of integer multiples of a single unit fraction, the inverse of their lowest common denominator, and generates as a subgroup a cyclic group of integer multiples of this unit fraction. A group is locally cyclic if and only if its lattice of subgroups is a distributive lattice.
Cyclically ordered groups.
A cyclically ordered group is a group together with a cyclic order preserved by the group structure. Every cyclic group can be given a structure as a cyclically ordered group, consistent with the ordering of the integers (or the integers modulo the order of the group).
Every finite subgroup of a cyclically ordered group is cyclic.
Metacyclic and polycyclic groups.
A metacyclic group is a group containing a cyclic normal subgroup whose quotient is also cyclic. These groups include the cyclic groups, the dicyclic groups, and the direct products of two cyclic groups. The polycyclic groups generalize metacyclic groups by allowing more than one level of group extension. A group is polycyclic if it has a finite descending sequence of subgroups, each of which is normal in the previous subgroup with a cyclic quotient, ending in the trivial group. Every finitely generated abelian group or nilpotent group is polycyclic.
Footnotes.
Notes.
Citations.
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The order of a subgroup of a finite group G divides the order of G
In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup $H$ of a finite group $G$, not only is $|G|/|H|$ an integer, but its value is the index $[G:H]$, defined as the number of left cosets of $H$ in $G$.
This variant holds even if $G$ is infinite, provided that $|G|$, $|H|$, and $[G:H]$ are interpreted as cardinal numbers.
Proof.
The left cosets of H in G are the equivalence classes of a certain equivalence relation on G: specifically, call x and y in G equivalent if there exists h in H such that "x"
"yh".
Therefore, the left cosets form a partition of G.
Each left coset "aH" has the same cardinality as H because $x \mapsto ax$ defines a bijection $H \to aH$ (the inverse is $y \mapsto a^{-1}y$).
The number of left cosets is the index ["G" : "H"].
By the previous three sentences,
$\left|G\right| = \left[G : H\right] \cdot \left|H\right|.$
Extension.
Lagrange's theorem can be extended to the equation of indexes between three subgroups of G.
If we take "K"
{"e"} (e is the identity element of G), then ["G" : {"e"}]
|"G"| and ["H" : {"e"}]
|"H"|. Therefore, we can recover the original equation |"G"|
["G" : "H"] |"H"|.
Applications.
A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer number k with , where e is the identity element of the group) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n elements, it follows
$\displaystyle a^n = e\mbox{.}$
This can be used to prove Fermat's little theorem and its generalization, Euler's theorem. These special cases were known long before the general theorem was proved.
The theorem also shows that any group of prime order is cyclic and simple, since the subgroup generated by any non-identity element must be the whole group itself.
Lagrange's theorem can also be used to show that there are infinitely many primes: suppose there were a largest prime $p$. Any prime divisor $q$ of the Mersenne number $2^p -1$ satisfies $2^p \equiv 1 \pmod {q}$ (see modular arithmetic), meaning that the order of $2$ in the multiplicative group $(\mathbb Z/q\mathbb Z)^*$ is $p$. By Lagrange's theorem, the order of $2$ must divide the order of $(\mathbb Z/q\mathbb Z)^*$, which is $q-1$. So $p$ divides $q-1$, giving $ p < q $, contradicting the assumption that $p$ is the largest prime.
Existence of subgroups of given order.
Lagrange's theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. This does not hold in general: given a finite group "G" and a divisor "d" of |"G"|, there does not necessarily exist a subgroup of "G" with order "d". The smallest example is "A"4 (the alternating group of degree 4), which has 12 elements but no subgroup of order 6.
A "Converse of Lagrange's Theorem" (CLT) group is a finite group with the property that for every divisor of the order of the group, there is a subgroup of that order. It is known that a CLT group must be solvable and that every supersolvable group is a CLT group. However, there exist solvable groups that are not CLT (for example, "A"4) and CLT groups that are not supersolvable (for example, "S"4, the symmetric group of degree 4).
There are partial converses to Lagrange's theorem. For general groups, Cauchy's theorem guarantees the existence of an element, and hence of a cyclic subgroup, of order any prime dividing the group order. Sylow's theorem extends this to the existence of a subgroup of order equal to the maximal power of any prime dividing the group order. For solvable groups, Hall's theorems assert the existence of a subgroup of order equal to any unitary divisor of the group order (that is, a divisor coprime to its cofactor).
Counterexample of the converse of Lagrange's theorem.
The converse of Lagrange's theorem states that if d is a divisor of the order of a group G, then there exists a subgroup H where |"H"| = "d".
We will examine the alternating group "A"4, the set of even permutations as the subgroup of the Symmetric group "S"4.
Let V be the non-cyclic subgroup of "A"4 called the Klein four-group.
"V" = {"e", (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}.
Let "K" = "H" ⋂ "V". Since both H and V are subgroups of "A"4, K is also a subgroup of "A"4.
From Lagrange's theorem, the order of K must divide both 6 and 4, the orders of H and V respectively. The only two positive integers that divide both 6 and 4 are 1 and 2. So |"K"| = 1 or 2.
Assume |"K"| = 1, then "K" = {"e"}. If H does not share any elements with V, then the 5 elements in H besides the Identity element e must be of the form ("a b c") where "a, b, c" are distinct elements in {1, 2, 3, 4}.
Since any element of the form ("a b c") squared is ("a c b"), and ("a b c")("a c b") = "e", any element of H in the form ("a b c") must be paired with its inverse. Specifically, the remaining 5 elements of H must come from distinct pairs of elements in "A"4 that are not in V. This is impossible since pairs of elements must be even and cannot total up to 5 elements. Thus, the assumptions that |"K"| = 1 is wrong, so |"K"| = 2.
Then, "K" = {"e", "v"} where "v" ∈ "V", v must be in the form ("a b")("c d") where a, b, c, d are distinct elements of {1, 2, 3, 4}. The other four elements in H are cycles of length 3.
Note that the cosets generated by a subgroup of a group form a partition of the group. The cosets generated by a specific subgroup are either identical to each other or disjoint. The index of a subgroup in a group ["A"4 : "H"] = |"A"4|/|"H"| is the number of cosets generated by that subgroup. Since |"A"4| = 12 and |"H"| = 6, H will generate two left cosets, one that is equal to H and another, gH, that is of length 6 and includes all the elements in "A"4 not in H.
Since there are only 2 distinct cosets generated by H, then H must be normal. Because of that, "H" = "gHg"−1 (∀"g" ∈ "A"4). In particular, this is true for "g" = ("a b c") ∈ "A"4. Since "H" = "gHg"−1, "gvg"−1 ∈ "H".
Without loss of generality, assume that "a" = 1, "b" = 2, "c" = 3, "d" = 4. Then "g" = (1 2 3), "v" = (1 2)(3 4), "g"−1 = (1 3 2), "gv" = (1 3 4), "gvg"−1 = (1 4)(2 3). Transforming back, we get "gvg"−1 = ("a" "d")("b" "c"). Because V contains all disjoint transpositions in "A"4, "gvg"−1 ∈ "V". Hence, "gvg"−1 ∈ "H" ⋂ "V" = "K".
Since "gvg"−1 ≠ "v", we have demonstrated that there is a third element in K. But earlier we assumed that |"K"| = 2, so we have a contradiction.
Therefore, our original assumption that there is a subgroup of order 6 is not true and consequently there is no subgroup of order 6 in "A"4 and the converse of Lagrange's theorem is not necessarily true.
Q.E.D.
History.
Lagrange himself did not prove the theorem in its general form. He stated, in his article "Réflexions sur la résolution algébrique des équations", that if a polynomial in n variables has its variables permuted in all "n"! ways, the number of different polynomials that are obtained is always a factor of "n"!. (For example, if the variables x, y, and z are permuted in all 6 possible ways in the polynomial "x" + "y" − "z" then we get a total of 3 different polynomials: "x" + "y" − "z", "x" + "z" − "y", and "y" + "z" − "x". Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group "S"n of the subgroup "H" of permutations that preserve the polynomial. (For the example of "x" + "y" − "z", the subgroup "H" in "S"3 contains the identity and the transposition ("x y").) So the size of "H" divides "n"!. With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.
In his "Disquisitiones Arithmeticae" in 1801, Carl Friedrich Gauss proved Lagrange's theorem for the special case of $(\mathbb Z/p \mathbb Z)^*$, the multiplicative group of nonzero integers modulo p, where p is a prime. In 1844, Augustin-Louis Cauchy proved Lagrange's theorem for the symmetric group "S"n.
Camille Jordan finally proved Lagrange's theorem for the case of any permutation group in 1861.
Notes.
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Representation of groups by permutations
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group.
More specifically, G is isomorphic to a subgroup of the symmetric group $\operatorname{Sym}(G)$ whose elements are the permutations of the underlying set of G.
Explicitly,
The homomorphism $G \to \operatorname{Sym}(G)$ can also be understood as arising from the left translation action of G on the underlying set G.
When G is finite, $\operatorname{Sym}(G)$ is finite too. The proof of Cayley's theorem in this case shows that if G is a finite group of order n, then G is isomorphic to a subgroup of the standard symmetric group $S_n$. But G might also be isomorphic to a subgroup of a smaller symmetric group, $S_m$ for some $m<n$; for instance, the order 6 group $G=S_3$ is not only isomorphic to a subgroup of $S_6$, but also (trivially) isomorphic to a subgroup of $S_3$. The problem of finding the minimal-order symmetric group into which a given group G embeds is rather difficult.
Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups".
When G is infinite, $\operatorname{Sym}(G)$ is infinite, but Cayley's theorem still applies.
History.
While it seems elementary enough, at the time the modern definitions did not exist, and when Cayley introduced what are now called "groups" it was not immediately clear that this was equivalent to the previously known groups, which are now called "permutation groups". Cayley's theorem unifies the two.
Although Burnside
attributes the theorem
to Jordan,
Eric Nummela
nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,
showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an embedding). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.
The theorem was later published by Walther Dyck in 1882 and is attributed to Dyck in the first edition of Burnside's book.
Background.
A "permutation" of a set A is a bijective function from A to A. The set of all permutations of A forms a group under function composition, called "the symmetric group on" A, and written as $\operatorname{Sym}(A)$.
In particular, taking A to be the underlying set of a group G produces a symmetric group denoted $\operatorname{Sym}(G)$.
Proof of the theorem.
If "g" is any element of a group "G" with operation ∗, consider the function "f""g" : "G" → "G", defined by "f""g"("x") = "g" ∗ "x". By the existence of inverses, this function has also an inverse, $f_{g^{-1}}$. So multiplication by "g" acts as a bijective function. Thus, "f""g" is a permutation of "G", and so is a member of Sym("G").
The set "K" = {"f""g" : "g" ∈ "G"} is a subgroup of Sym("G") that is isomorphic to "G". The fastest way to establish this is to consider the function "T" : "G" → Sym("G") with "T"("g") = "f""g" for every "g" in "G". "T" is a group homomorphism because (using · to denote composition in Sym("G")):
$ (f_g \cdot f_h)(x) = f_g(f_h(x)) = f_g(h*x) = g*(h*x) = (g*h)*x = f_{g*h}(x) ,$
for all "x" in "G", and hence:
$ T(g) \cdot T(h) = f_g \cdot f_h = f_{g*h} = T(g*h) .$
The homomorphism "T" is injective since "T"("g") = id"G" (the identity element of Sym("G")) implies that "g" ∗ "x" = "x" for all "x" in "G", and taking "x" to be the identity element "e" of "G" yields "g" = "g" ∗ "e" = "e", i.e. the kernel is trivial. Alternatively, "T" is also injective since "g" ∗ "x" = "g"′ ∗ "x" implies that "g" = "g"′ (because every group is cancellative).
Thus "G" is isomorphic to the image of "T", which is the subgroup "K".
"T" is sometimes called the "regular representation of" "G".
Alternative setting of proof.
An alternative setting uses the language of group actions. We consider the group $G$ as acting on itself by left multiplication, i.e. $g \cdot x = gx$, which has a permutation representation, say $\phi : G \to \mathrm{Sym}(G)$.
The representation is faithful if $\phi$ is injective, that is, if the kernel of $\phi$ is trivial. Suppose $g\in\ker\phi$. Then, $g = ge = g\cdot e = e$. Thus, $\ker\phi$ is trivial. The result follows by use of the first isomorphism theorem, from which we get $\mathrm{Im}\, \phi \cong G$.
Remarks on the regular group representation.
The identity element of the group corresponds to the identity permutation. All other group elements correspond to derangements: permutations that do not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation that consists of cycles all of the same length: this length is the order of that element. The elements in each cycle form a right coset of the subgroup generated by the element.
Examples of the regular group representation.
Z2 = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12) (see cycle notation). E.g. 0 +1 = 1 and 1+1 = 0, so <math display=inline>1\mapsto0$ and <math display=inline>0\mapsto1,$ as they would under a permutation.
$ \mathbb Z_3 = \{0,1,2\} $ with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132). E.g. 1 + 1 = 2 corresponds to (123)(123) = (132).
Z4 = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).
The elements of Klein four-group {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).
S3 (dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements, and the latter is how it is realized by its regular representation.
More general statement.
Theorem:
Let G be a group, and let H be a subgroup.
Let $G/H$ be the set of left cosets of H in G.
Let N be the normal core of H in G, defined to be the intersection of the conjugates of H in G.
Then the quotient group $G/N$ is isomorphic to a subgroup of $\operatorname{Sym}(G/H)$.
The special case $H=1$ is Cayley's original theorem.
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Group that can be constructed from abelian groups using extensions
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.
Motivation.
Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial $f \in F[x]$ there is a tower of field extensions$F = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=K$such that
Example.
For example, the smallest Galois field extension of $\mathbb{Q}$ containing the element$a = \sqrt[5]{\sqrt{2} + \sqrt{3}}$gives a solvable group. It has associated field extensions$\mathbb{Q}
\subseteq \mathbb{Q}(\sqrt{2})
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}, a\right)$giving a solvable group of Galois extensions containing the following composition factors:
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)/
\mathbb{Q}(\sqrt{2}, \sqrt{3})
\right)
\cong \mathbb{Z}/4 $ with group action $h^n\left(e^{2im\pi/5}\right) = e^{2(n+1)mi\pi/5} , \ 0 \leq n \leq 3, \ h^4 = 1$, and minimal polynomial $x^4 + x^3+x^2+x+1 = (x^5 - 1)/(x-1)$ containing the 5th roots of unity excluding $1$.
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}, a\right)/
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)
\right)
\cong \mathbb{Z}/5 $ with group action $j^l(a) = e^{2li\pi/5}a, \ j^5 = 1$, and minimal polynomial $x^5 - \left(\sqrt{2} + \sqrt{3}\right)$.
, where $1$ is the identity permutation. All of the defining group actions change a single extension while keeping all of the other extensions fixed. For example, an element of this group is the group action $fgh^3j^4 $. A general element in the group can be written as $f^ag^bh^nj^l,\ 0 \leq a, b \leq 1,\ 0 \leq n \leq 3,\ 0 \leq l \leq 4 $ for a total of 80 elements.
It is worthwhile to note that this group is not abelian itself. For example:
$hj(a) = h(e^{2i\pi/5}a) = e^{4i\pi/5}a $
$jh(a) = j(a) = e^{2i\pi/5}a $
In fact, in this group, $jh = hj^3 $. The solvable group is isometric to $(\mathbb{C}_5 \rtimes_\varphi \mathbb{C}_4) \times (\mathbb{C}_2 \times \mathbb{C}_2),\ \mathrm{where}\ \varphi_h(j) = hjh^{-1} = j^2 $, defined using the semidirect product and direct product of the cyclic groups. In the solvable group, $\mathbb{C}_4 $ is not a normal subgroup.
Definition.
A group "G" is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups 1 = "G"0 < "G"1 < ⋅⋅⋅ < "Gk" = "G" such that "G""j"−1 is normal in "Gj", and "Gj "/"G""j"−1 is an abelian group, for "j" = 1, 2, …, "k".
Or equivalently, if its derived series, the descending normal series
$G\triangleright G^{(1)}\triangleright G^{(2)} \triangleright \cdots,$
where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup of "G". These two definitions are equivalent, since for every group "H" and every normal subgroup "N" of "H", the quotient "H"/"N" is abelian if and only if "N" includes the commutator subgroup of "H". The least "n" such that "G"("n") = 1 is called the derived length of the solvable group "G".
For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, and every simple abelian group is cyclic of prime order. The Jordan–Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to "n"th roots (radicals) over some field. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series {0, Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.
Examples.
Abelian groups.
The basic example of solvable groups are abelian groups. They are trivially solvable since a subnormal series is formed by just the group itself and the trivial group. But non-abelian groups may or may not be solvable.
Nilpotent groups.
More generally, all nilpotent groups are solvable. In particular, finite "p"-groups are solvable, as all finite "p"-groups are nilpotent.
Quaternion groups.
In particular, the quaternion group is a solvable group given by the group extension$1 \to \mathbb{Z}/2 \to Q \to \mathbb{Z}/2 \times \mathbb{Z}/2 \to 1$where the kernel $\mathbb{Z}/2$ is the subgroup generated by $-1$.
Group extensions.
Group extensions form the prototypical examples of solvable groups. That is, if $G$ and $G'$ are solvable groups, then any extension$1 \to G \to G" \to G' \to 1$defines a solvable group $G"$. In fact, all solvable groups can be formed from such group extensions.
Non-abelian group which is non-nilpotent.
A small example of a solvable, non-nilpotent group is the symmetric group "S"3. In fact, as the smallest simple non-abelian group is "A"5, (the alternating group of degree 5) it follows that "every" group with order less than 60 is solvable.
Finite groups of odd order.
The Feit–Thompson theorem states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.
Non-example.
The group "S"5 is not solvable — it has a composition series {E, "A"5, "S"5} (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to "A"5 and "C"2; and "A"5 is not abelian. Generalizing this argument, coupled with the fact that "A""n" is a normal, maximal, non-abelian simple subgroup of "S""n" for "n" > 4, we see that "S""n" is not solvable for "n" > 4. This is a key step in the proof that for every "n" > 4 there are polynomials of degree "n" which are not solvable by radicals (Abel–Ruffini theorem). This property is also used in complexity theory in the proof of Barrington's theorem.
Subgroups of GL2.
0 & *
1 & * \\
0 & 1
a & b \\
0 & c
\end{bmatrix} \cdot
1 & d \\
0 & 1
\end{bmatrix} =
a & ad + b \\
0 & c
$Note the determinant condition on $GL_2
$ implies $ac \neq 0
$, hence $\mathbb{F}^\times \times \mathbb{F}^\times \subset B
$ is a subgroup (which are the matrices where $b=0
$). For fixed $a,b
$, the linear equation $ad + b = 0
$ implies $d = -b/a
$. Since we can take any matrix in $B
1 & d \\
0 & 1
$with $d = -b/a
$, we can get a diagonal matrix in $B
$. This shows the quotient group $B/U \cong \mathbb{F}^\times \times \mathbb{F}^\times$.
Remark.
Notice that this description gives the decomposition of $B
$ as $\mathbb{F} \rtimes (\mathbb{F}^\times \times \mathbb{F}^\times)
$ where $(a,c)
$ acts on $b
$ by $(a,c)(b) = ab
$. This implies $(a,c)(b + b') = (a,c)(b) + (a,c)(b') = ab + ab'
a & b \\
0 & c
\end{bmatrix}$corresponds to the element $(b) \times (a,c)$ in the group.
Borel subgroups.
For a linear algebraic group $G$ its Borel subgroup is defined as a subgroup which is closed, connected, and solvable in $G$, and it is the maximal possible subgroup with these properties (note the second two are topological properties). For example, in $GL_n$ and $SL_n$ the group of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup $B$ in $GL_2$ is the Borel subgroup.
Borel subgroup in GL3.
In $GL_3$ there are the subgroups$B = \left\{
0 & * & * \\
0 & 0 & *
U_1 = \left\{
1 & * & * \\
0 & 1 & * \\
0 & 0 & 1
\right\}$Notice $B/U_1 \cong \mathbb{F}^\times \times \mathbb{F}^\times \times \mathbb{F}^\times$, hence the Borel group has the form$U\rtimes
$
Borel subgroup in product of simple linear algebraic groups.
T & 0 \\
0 & S
\end{bmatrix}$where $T$ is an $n\times n$ upper triangular matrix and $S$ is a $m\times m$ upper triangular matrix.
Z-groups.
Any finite group whose "p"-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable. Such groups are called Z-groups.
OEIS values.
Numbers of solvable groups with order "n" are (start with "n" = 0)
0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, ... (sequence in the OEIS)
Orders of non-solvable groups are
60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, ... (sequence in the OEIS)
Properties.
Solvability is closed under a number of operations.
Solvability is closed under group extension:
It is also closed under wreath product:
For any positive integer "N", the solvable groups of derived length at most "N" form a subvariety of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras, and (direct) products. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.
Burnside's theorem.
Burnside's theorem states that if "G" is a finite group of order "paqb" where "p" and "q" are prime numbers, and "a" and "b" are non-negative integers, then "G" is solvable.
Related concepts.
Supersolvable groups.
As a strengthening of solvability, a group "G" is called supersolvable (or supersoluble) if it has an "invariant" normal series whose factors are all cyclic. Since a normal series has finite length by definition, uncountable groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group "A"4 is an example of a finite solvable group that is not supersolvable.
If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:
cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group.
Virtually solvable groups.
A group "G" is called virtually solvable if it has a solvable subgroup of finite index. This is similar to virtually abelian. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.
Hypoabelian.
A solvable group is one whose derived series reaches the trivial subgroup at a "finite" stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group. The first ordinal "α" such that "G"("α") = "G"("α"+1) is called the (transfinite) derived length of the group "G", and it has been shown that every ordinal is the derived length of some group .
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Commutative group where every element is the sum of elements from one finite subset
In abstract algebra, an abelian group $(G,+)$ is called finitely generated if there exist finitely many elements $x_1,\dots,x_s$ in $G$ such that every $x$ in $G$ can be written in the form $x = n_1x_1 + n_2x_2 + \cdots + n_sx_s$ for some integers $n_1,\dots, n_s$. In this case, we say that the set $\{x_1,\dots, x_s\}$ is a "generating set" of $G$ or that $x_1,\dots, x_s$ "generate" $G$.
Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.
Examples.
There are no other examples (up to isomorphism). In particular, the group $\left(\mathbb{Q},+\right)$ of rational numbers is not finitely generated: if $x_1,\ldots,x_n$ are rational numbers, pick a natural number $k$ coprime to all the denominators; then $1/k$ cannot be generated by $x_1,\ldots,x_n$. The group $\left(\mathbb{Q}^*,\cdot\right)$ of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition $ \left(\mathbb{R},+\right)$ and non-zero real numbers under multiplication $\left(\mathbb{R}^*,\cdot\right)$ are also not finitely generated.
Classification.
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of "finite" abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
Primary decomposition.
The primary decomposition formulation states that every finitely generated abelian group "G" is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups. A primary cyclic group is one whose order is a power of a prime. That is, every finitely generated abelian group is isomorphic to a group of the form
$\mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z},$
where "n" ≥ 0 is the "rank", and the numbers "q"1, ..., "q""t" are powers of (not necessarily distinct) prime numbers. In particular, "G" is finite if and only if "n" = 0. The values of "n", "q"1, ..., "q""t" are (up to rearranging the indices) uniquely determined by "G", that is, there is one and only one way to represent "G" as such a decomposition.
The proof of this statement uses the basis theorem for finite abelian group: every finite abelian group is a direct sum of primary cyclic groups. Denote the torsion subgroup of "G" as "tG". Then, "G/tG" is a torsion-free abelian group and thus it is free abelian. "tG" is a direct summand of "G", which means there exists a subgroup "F" of "G" s.t. $G=tG\oplus F$, where $F\cong G/tG$. Then, "F" is also free abelian. Since "tG" is finitely generated and each element of "tG" has finite order, "tG" is finite. By the basis theorem for finite abelian group, "tG" can be written as direct sum of primary cyclic groups.
Invariant factor decomposition.
We can also write any finitely generated abelian group "G" as a direct sum of the form
$\mathbb{Z}^n \oplus \mathbb{Z}/{k_1}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/{k_u}\mathbb{Z},$
where "k"1 divides "k"2, which divides "k"3 and so on up to "k""u". Again, the rank "n" and the "invariant factors" "k"1, ..., "k""u" are uniquely determined by "G" (here with a unique order). The rank and the sequence of invariant factors determine the group up to isomorphism.
Equivalence.
These statements are equivalent as a result of the Chinese remainder theorem, which implies that $\mathbb{Z}_{jk}\cong \mathbb{Z}_{j} \oplus \mathbb{Z}_{k}$ if and only if "j" and "k" are coprime.
History.
The history and credit for the fundamental theorem is complicated by the fact that it was proven when group theory was not well-established, and thus early forms, while essentially the modern result and proof, are often stated for a specific case. Briefly, an early form of the finite case was proven by Gauss in 1801, the finite case was proven by Kronecker in 1870, and stated in group-theoretic terms by Frobenius and Stickelberger in 1878. The finitely "presented" case is solved by Smith normal form, and hence frequently credited to , though the finitely "generated" case is sometimes instead credited to Poincaré in 1900; details follow.
Group theorist László Fuchs states:
As far as the fundamental theorem on finite abelian groups is concerned, it is not clear how far back in time one needs to go to trace its origin. ... it took a long time to formulate and prove the fundamental theorem in its present form ...
The fundamental theorem for "finite" abelian groups was proven by Leopold Kronecker in 1870, using a group-theoretic proof, though without stating it in group-theoretic terms; a modern presentation of Kronecker's proof is given in , 5.2.2 Kronecker's Theorem, 176–177. This generalized an earlier result of Carl Friedrich Gauss from "Disquisitiones Arithmeticae" (1801), which classified quadratic forms; Kronecker cited this result of Gauss's. The theorem was stated and proved in the language of groups by Ferdinand Georg Frobenius and Ludwig Stickelberger in 1878. Another group-theoretic formulation was given by Kronecker's student Eugen Netto in 1882.
The fundamental theorem for "finitely presented" abelian groups was proven by Henry John Stephen Smith in , as integer matrices correspond to finite presentations of abelian groups (this generalizes to finitely presented modules over a principal ideal domain), and Smith normal form corresponds to classifying finitely presented abelian groups.
The fundamental theorem for "finitely generated" abelian groups was proven by Henri Poincaré in 1900, using a matrix proof (which generalizes to principal ideal domains). This was done in the context of computing the
homology of a complex, specifically the Betti number and torsion coefficients of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part.
Kronecker's proof was generalized to "finitely generated" abelian groups by Emmy Noether in 1926.
Corollaries.
Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of "G". The rank of "G" is defined as the rank of the torsion-free part of "G"; this is just the number "n" in the above formulas.
A corollary to the fundamental theorem is that every finitely generated torsion-free abelian group is free abelian. The finitely generated condition is essential here: $\mathbb{Q}$ is torsion-free but not free abelian.
Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the group homomorphisms, form an abelian category which is a Serre subcategory of the category of abelian groups.
Non-finitely generated abelian groups.
Note that not every abelian group of finite rank is finitely generated; the rank 1 group $\mathbb{Q}$ is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of $\mathbb{Z}_{2}$ is another one.
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In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.
Statement.
Let "A" be an abelian group. If "A" is finitely generated then by the fundamental theorem of finitely generated abelian groups, "A" is decomposable into a direct sum of cyclic subgroups, which leads to the classification of finitely generated abelian groups up to isomorphism. The structure of general infinite abelian groups can be considerably more complicated and the conclusion needs not to hold, but Prüfer proved that it remains true for periodic groups in two special cases.
The first Prüfer theorem states that an abelian group of bounded exponent is isomorphic to a direct sum of cyclic groups. The second Prüfer theorem states that a countable abelian "p"-group whose non-trivial elements have finite "p"-height is isomorphic to a direct sum of cyclic groups. Examples show that the assumption that the group be countable cannot be removed.
The two Prüfer theorems follow from a general criterion of decomposability of an abelian group into a direct sum of cyclic subgroups due to L. Ya. Kulikov:
An abelian "p"-group "A" is isomorphic to a direct sum of cyclic groups if and only if it is a union of a sequence {"A""i"} of subgroups with the property that the heights of all elements of "A""i" are bounded by a constant (possibly depending on "i").
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In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term "quasinormal subgroup" was introduced by Øystein Ore in 1937.
Two subgroups are said to permute (or commute) if any element from the first
subgroup, times an element of the second subgroup, can be written as an element of the second
subgroup, times an element of the first subgroup. That is, $H$ and $K$
as subgroups of $G$ are said to commute if "HK" = "KH", that is, any element of the form $hk$
with $h \in H$ and $k \in K$ can be written in the form $k'h'$
where $k' \in K$ and $h' \in H$.
Every normal subgroup is quasinormal, because a normal subgroup commutes with every element of the group. The converse is not true. For instance, any extension of a cyclic $p$-group by another cyclic $p$-group for the same (odd) prime has the property that all its subgroups are quasinormal. However, not all of its subgroups need be normal.
Every quasinormal subgroup is a modular subgroup, that is, a modular element in the lattice of subgroups. This follows from the modular property of groups. If all subgroups are quasinormal, then the group is called an Iwasawa group—sometimes also called a "modular group", although this latter term has other meanings.
In any group, every quasinormal subgroup is ascendant.
A conjugate permutable subgroup is one that commutes with all its conjugate subgroups. Every quasinormal subgroup is conjugate permutable.
In finite groups.
Every quasinormal subgroup of a finite group is a subnormal subgroup. This follows from the somewhat stronger statement that every conjugate permutable subgroup is subnormal, which in turn follows from the statement that every maximal conjugate permutable subgroup is normal. (The finiteness is used crucially in the proofs.)
In summary, a subgroup "H" of a finite group "G" is permutable in "G" if and only if "H" is both modular and subnormal in "G".
PT-groups.
Permutability is not a transitive relation in general. The groups in which permutability is transitive are called PT-groups, by analogy with T-groups in which normality is transitive.
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Existence of group elements of prime order
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with xp = e, where e is the identity element of G. It is named after Augustin-Louis Cauchy, who discovered it in 1845.
The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group G divides the order of G. Cauchy's theorem implies that for any prime divisor p of the order of G, there is a subgroup of G whose order is p—the cyclic group generated by the element in Cauchy's theorem.
Cauchy's theorem is generalized by Sylow's first theorem, which implies that if pn is the maximal power of p dividing the order of G, then G has a subgroup of order pn (and using the fact that a p-group is solvable, one can show that G has subgroups of order pr for any r less than or equal to n).
Statement and proof.
Many texts prove the theorem with the use of strong induction and the class equation, though considerably less machinery is required to prove the theorem in the abelian case. One can also invoke group actions for the proof.
Theorem — Cauchy's theorem
Proof 1.
We first prove the special case that where G is abelian, and then the general case; both proofs are by induction on n = |G|, and have as starting case n = p which is trivial because any non-identity element now has order p. Suppose first that G is abelian. Take any non-identity element a, and let H be the cyclic group it generates. If p divides |H|, then a|H|/p is an element of order p. If p does not divide |H|, then it divides the order [G:H] of the quotient group G/H, which therefore contains an element of order p by the inductive hypothesis. That element is a class xH for some x in G, and if m is the order of x in G, then xm = e in G gives (xH)m = eH in G/H, so p divides m; as before xm/p is now an element of order p in G, completing the proof for the abelian case.
In the general case, let Z be the center of G, which is an abelian subgroup. If p divides |Z|, then Z contains an element of order p by the case of abelian groups, and this element works for G as well. So we may assume that p does not divide the order of Z. Since p does divide |G|, and G is the disjoint union of Z and of the conjugacy classes of non-central elements, there exists a conjugacy class of a non-central element a whose size is not divisible by p. But the class equation shows that size is [G : CG(a)], so p divides the order of the centralizer CG(a) of a in G, which is a proper subgroup because a is not central. This subgroup contains an element of order p by the inductive hypothesis, and we are done.
Proof 2.
This proof uses the fact that for any action of a (cyclic) group of prime order p, the only possible orbit sizes are 1 and p, which is immediate from the orbit stabilizer theorem.
The set that our cyclic group shall act on is the set
$ X = \{\,(x_1,\ldots,x_p) \in G^p : x_1x_2\cdots x_p = e\, \} $
of p-tuples of elements of G whose product (in order) gives the identity. Such a p-tuple is uniquely determined by all its components except the last one, as the last element must be the inverse of the product of those preceding elements. One also sees that those elements can be chosen freely, so X has |G|p−1 elements, which is divisible by p.
Now from the fact that in a group if ab = e then also ba = e, it follows that any cyclic permutation of the components of an element of X again gives an element of X. Therefore one can define an action of the cyclic group Cp of order p on X by cyclic permutations of components, in other words in which a chosen generator of Cp sends
$(x_1,x_2,\ldots,x_p)\mapsto(x_2,\ldots,x_p,x_1)$.
As remarked, orbits in X under this action either have size 1 or size p. The former happens precisely for those tuples $(x,x,\ldots,x)$ for which $x^p=e$. Counting the elements of X by orbits, and reducing modulo p, one sees that the number of elements satisfying $x^p=e$ is divisible by p. But x = e is one such element, so there must be at least other solutions for x, and these solutions are elements of order p. This completes the proof.
Uses.
A practically immediate consequence of Cauchy's theorem is a useful characterization of finite p-groups, where p is a prime. In particular, a finite group G is a p-group (i.e. all of its elements have order pk for some natural number k) if and only if G has order pn for some natural number n. One may use the abelian case of Cauchy's Theorem in an inductive proof of the first of Sylow's theorems, similar to the first proof above, although there are also proofs that avoid doing this special case separately.
Example 1.
Let G be a finite group where for all elements x of G. Then G has the order for some non negative integer n. Let equal m. In the case of m is 1, then . In the case of , if m has the odd prime factor p, G has the element x where from Cauchy's theorem. It conflicts with the assumption. Therefore m must be . G is an abelian group, and G is called an elementary abelian 2-group or Boolean group. The well-known example is Klein four-group.
Example 2.
An abelian simple group is either {"e"} or cyclic group Cp whose order is a prime number p. Let G is an abelian group, then all subgroups of G are normal subgroups. So, if G is a simple group, G has only normal subgroup that is either {"e"} or G. If , then G is {"e"}. It is suitable. If , let "a" ∈ "G" is not e, the cyclic group $\langle a \rangle$ is subgroup of G and $\langle a \rangle$ is not {"e"}, then $G = \langle a \rangle.$ Let n is the order of $\langle a \rangle$. If n is infinite, then
$G = \langle a \rangle \supsetneqq \langle a^2 \rangle \supsetneqq \{e\}.$
So in this case, it is not suitable. Then n is finite. If n is composite, n is divisible by prime q which is less than n. From Cauchy's theorem, the subgroup H will be exist whose order is q, it is not suitable. Therefore, n must be a prime number.
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In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and fusion such as described in . Various applications of these ideas include local criteria for "p"-nilpotence and various non-simplicity criteria focussing on showing that a finite group has a normal subgroup of index "p".
Background.
The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of "p", the transfer homomorphism, and fusion of elements.
Subgroups.
The following three normal subgroups of index a power of "p" are naturally defined, and arise as the smallest normal subgroups such that the quotient is (a certain kind of) "p"-group. Formally, they are kernels of the reflection onto the reflective subcategory of "p"-groups (respectively, elementary abelian "p"-groups, abelian "p"-groups).
Firstly, as these are weaker conditions on the groups "K," one obtains the containments $\mathbf{E}^p(G) \supseteq \mathbf{A}^p(G) \supseteq \mathbf{O}^p(G).$ These are further related as:
A"p"("G") = O"p"("G")["G","G"].
O"p"("G") has the following alternative characterization as the subgroup generated by all Sylow "q"-subgroups of "G" as "q"≠"p" ranges over the prime divisors of the order of "G" distinct from "p".
O"p"("G") is used to define the lower "p"-series of "G", similarly to the upper "p"-series described in p-core.
Transfer homomorphism.
The transfer homomorphism is a homomorphism that can be defined from any group "G" to the abelian group "H"/["H","H"] defined by a subgroup "H" ≤ "G" of finite index, that is ["G":"H"] < ∞. The transfer map from a finite group "G" into its Sylow "p"-subgroup has a kernel that is easy to describe:
The kernel of the transfer homomorphism from a finite group "G" into its Sylow "p"-subgroup "P" has A"p"("G") as its kernel, .
In other words, the "obvious" homomorphism onto an abelian "p"-group is in fact the most general such homomorphism.
Fusion.
The fusion pattern of a subgroup "H" in "G" is the equivalence relation on the elements of "H" where two elements "h", "k" of "H" are fused if they are "G"-conjugate, that is, if there is some "g" in "G" such that "h" = "k""g". The normal structure of "G" has an effect on the fusion pattern of its Sylow "p"-subgroups, and conversely the fusion pattern of its Sylow "p"-subgroups has an effect on the normal structure of "G", .
Focal subgroup.
One can define, as in the focal subgroup of "H" with respect to "G" as:
Foc"G"("H") = ⟨ "x"−1 "y" | "x","y" in "H" and "x" is "G"-conjugate to "y" ⟩.
This focal subgroup measures the extent to which elements of "H" fuse in "G", while the previous definition measured certain abelian "p"-group homomorphic images of the group "G". The content of the focal subgroup theorem is that these two definitions of focal subgroup are compatible.
shows that the focal subgroup of "P" in "G" is the intersection "P"∩["G","G"] of the Sylow "p"-subgroup "P" of the finite group "G" with the derived subgroup ["G","G"] of "G". The focal subgroup is important as it is a Sylow "p"-subgroup of the derived subgroup. One also gets the following result:
There exists a normal subgroup "K" of "G" with "G"/"K" an abelian "p"-group isomorphic to "P"/"P"∩["G","G"] (here "K" denotes A"p"("G")), and
if "K" is a normal subgroup of "G" with "G"/"K" an abelian p-group, then "P"∩["G","G"] ≤ "K", and "G"/"K" is a homomorphic image of "P"/"P"∩["G","G"], .
Statement of the theorem.
The focal subgroup of a finite group "G" with Sylow "p"-subgroup "P" is given by:
"P"∩["G","G"] = "P"∩A"p"("G") = "P"∩ker("v") = Foc"G"("P") = ⟨ "x"−1 "y" | "x","y" in "P" and "x" is "G"-conjugate to "y" ⟩
where "v" is the transfer homomorphism from "G" to "P"/["P","P"], .
History and generalizations.
This connection between transfer and fusion is credited to , where, in different language, the focal subgroup theorem was proved along with various generalizations. The requirement that "G"/"K" be abelian was dropped, so that Higman also studied O"p"("G") and the nilpotent residual γ∞("G"), as so called hyperfocal subgroups. Higman also did not restrict to a single prime "p", but rather allowed "π"-groups for sets of primes "π" and used Philip Hall's theorem of Hall subgroups in order to prove similar results about the transfer into Hall "π"-subgroups; taking "π" = {"p"} a Hall "π"-subgroup is a Sylow "p"-subgroup, and the results of Higman are as presented above.
Interest in the hyperfocal subgroups was renewed by work of in understanding the modular representation theory of certain well behaved blocks. The hyperfocal subgroup of "P" in "G" can defined as "P"∩γ∞("G") that is, as a Sylow "p"-subgroup of the nilpotent residual of "G". If "P" is a Sylow "p"-subgroup of the finite group "G", then one gets the standard focal subgroup theorem:
"P"∩γ∞("G") = "P"∩O"p"("G") = ⟨ "x"−1 "y" : "x","y" in "P" and "y" = "x""g" for some "g" in "G" of order coprime to "p" ⟩
and the local characterization:
"P"∩O"p"("G") = ⟨ "x"−1 "y" : "x","y" in "Q" ≤ "P" and "y" = "x""g" for some "g" in N"G"("Q") of order coprime to "p" ⟩.
This compares to the local characterization of the focal subgroup as:
"P"∩A"p"("G") = ⟨ "x"−1 "y" : "x","y" in "Q" ≤ "P" and "y" = "x""g" for some "g" in N"G"("Q") ⟩.
Puig is interested in the generalization of this situation to fusion systems, a categorical model of the fusion pattern of a Sylow "p"-subgroup with respect to a finite group that also models the fusion pattern of a defect group of a "p"-block in modular representation theory. In fact fusion systems have found a number of surprising applications and inspirations in the area of algebraic topology known as equivariant homotopy theory. Some of the major algebraic theorems in this area only have topological proofs at the moment.
Other characterizations.
Various mathematicians have presented methods to calculate the focal subgroup from smaller groups. For instance, the influential work develops the idea of a local control of fusion, and as an example application shows that:
"P" ∩ A"p"("G") is generated by the commutator subgroups ["Q", N"G"("Q")] where "Q" varies over a family "C" of subgroups of "P"
The choice of the family "C" can be made in many ways ("C" is what is called a "weak conjugation family" in ), and several examples are given: one can take "C" to be all non-identity subgroups of "P", or the smaller choice of just the intersections "Q" = "P" ∩ "P""g" for "g" in "G" in which N"P"("Q") and N"P""g"("Q") are both Sylow "p"-subgroups of N"G"("Q"). The latter choice is made in . The work of studied aspects of the transfer and fusion as well, resulting in Grün's first theorem:
"P" ∩ A"p"("G") is generated by "P" ∩ ["N", "N"] and "P" ∩ ["Q", "Q"] where "N" = N"G"("P") and "Q" ranges over the set of Sylow "p"-subgroups "Q" = "P""g" of "G" .
Applications.
The textbook presentations in , , , , all contain various applications of the focal subgroup theorem relating fusion, transfer, and a certain kind of splitting called "p"-nilpotence.
During the course of the Alperin–Brauer–Gorenstein theorem classifying finite simple groups with quasi-dihedral Sylow 2-subgroups, it becomes necessary to distinguish four types of groups with quasi-dihedral Sylow 2-subgroups: the 2-nilpotent groups, the "Q"-type groups whose focal subgroup is a generalized quaternion group of index 2, the "D"-type groups whose focal subgroup a dihedral group of index 2, and the "QD"-type groups whose focal subgroup is the entire quasi-dihedral group. In terms of fusion, the 2-nilpotent groups have 2 classes of involutions, and 2 classes of cyclic subgroups of order 4; the "Q"-type have 2 classes of involutions and one class of cyclic subgroup of order 4; the "QD"-type have one class each of involutions and cyclic subgroups of order 4. In other words, finite groups with quasi-dihedral Sylow 2-subgroups can be classified according to their focal subgroup, or equivalently, according to their fusion patterns. The explicit lists of groups with each fusion pattern are contained in .
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2947865
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abstract_algebra
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Classification theorem in group theory
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963).
History.
The contrast that these results show between groups of odd and even order suggests inevitably that simple groups of odd order do not exist.
William Burnside (1911, p. 503 note M)
William Burnside (1911, p. 503 note M) conjectured that every nonabelian finite simple group has even order. Richard Brauer (1957) suggested using the centralizers of involutions of simple groups as the basis for the classification of finite simple groups, as the Brauer–Fowler theorem shows that there are only a finite number of finite simple groups with given centralizer of an involution. A group of odd order has no involutions, so to carry out Brauer's program it is first necessary to show that non-cyclic finite simple groups never have odd order. This is equivalent to showing that odd order groups are solvable, which is what Feit and Thompson proved.
The attack on Burnside's conjecture was started by Michio Suzuki (1957), who studied CA groups; these are groups such that the Centralizer of every non-trivial element is Abelian. In a pioneering paper he showed that all CA groups of odd order are solvable. (He later classified all the simple CA groups, and more generally all simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the process, that are now called Suzuki groups.)
Feit, Thompson, and Marshall Hall (1960) extended Suzuki's work to the family of CN groups; these are groups such that the Centralizer of every non-trivial element is Nilpotent. They showed that every CN group of odd order is solvable. Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long for a proof in group theory.
The Feit–Thompson theorem can be thought of as the next step in this process: they show that there is no non-cyclic simple group of odd order such that every proper subgroup is solvable. This proves that every finite group of odd order is solvable, as a minimal counterexample must be a simple group such that every proper subgroup is solvable. Although the proof follows the same general outline as the CA theorem and the CN theorem, the details are vastly more complicated. The final paper is 255 pages long.
Significance of the proof.
The Feit–Thompson theorem showed that the classification of finite simple groups using centralizers of involutions might be possible, as every nonabelian simple group has an involution. Many of the techniques they introduced in their proof, especially the idea of local analysis, were developed further into tools used in the classification. Perhaps the most revolutionary aspect of the proof was its length: before the Feit–Thompson paper, few arguments in group theory were more than a few pages long and most could be read in a day. Once group theorists realized that such long arguments could work, a series of papers that were several hundred pages long started to appear. Some of these dwarfed even the Feit–Thompson paper; the paper by Michael Aschbacher and Stephen D. Smith on quasithin groups was 1,221 pages long.
Revision of the proof.
Many mathematicians have simplified parts of the original Feit–Thompson proof. However all of these improvements are in some sense local; the global structure of the argument is still the same, but some of the details of the arguments have been simplified.
The simplified proof has been published in two books: , which covers everything except the character theory, and , which covers the character theory. This revised proof is still very hard, and is longer than the original proof, but is written in a more leisurely style.
A fully formal proof, checked with the Coq proof assistant, was announced in September 2012 by Georges Gonthier and fellow researchers at Microsoft Research and Inria.
An outline of the proof.
Instead of describing the Feit–Thompson theorem directly, it is easier to describe Suzuki's CA theorem and then comment on some of the extensions needed for the CN-theorem and the odd order theorem. The proof can be broken up into three steps. We let "G" be a non-abelian (minimal) simple group of odd order satisfying the CA condition. For a more detailed exposition of the odd order paper see or or .
Step 1. Local analysis of the structure of the group "G".
This is easy in the CA case because the relation ""a" commutes with "b"" is an equivalence relation on the non-identity elements. So the elements break up into equivalence classes, such that each equivalence class is the set of non-identity elements of a maximal abelian subgroup. The normalizers of these maximal abelian subgroups turn out to be exactly the maximal proper subgroups of "G". These normalizers are Frobenius groups whose character theory is reasonably transparent, and well-suited to manipulations involving character induction. Also, the set of prime divisors of |"G"| is partitioned according to the primes which divide the orders of the distinct conjugacy classes of maximal abelian subgroups of |"G"|. This pattern of partitioning the prime divisors of |"G"| according to conjugacy classes of certain Hall subgroups (a Hall subgroup is one whose order and index are relatively prime) which correspond to the maximal subgroups of "G" (up to conjugacy) is repeated in both the proof of the Feit–Hall–Thompson CN-theorem and in the proof of the Feit–Thompson odd-order theorem. Each maximal subgroup "M" has a certain nilpotent Hall subgroup "M"σ with normalizer contained in "M", whose order is divisible by certain primes forming a set σ("M"). Two maximal subgroups are conjugate if and only if the sets σ("M") are the same, and if they are not conjugate then the sets σ("M") are disjoint. Every prime dividing the order of "G" occurs in some set σ("M"). So the primes dividing the order of "G" are partitioned into equivalence classes corresponding to the conjugacy classes of maximal subgroups. The proof of the CN-case is already considerably more difficult than the CA-case: the main extra problem is to prove that two different Sylow subgroups intersect in the identity. This part of the proof of the odd-order theorem takes over 100 journal pages. A key step is the proof of the Thompson uniqueness theorem, stating that abelian subgroups of normal rank at least 3 are contained in a unique maximal subgroup, which means that the primes "p" for which the Sylow "p"-subgroups have normal rank at most 2 need to be considered separately. Bender later simplified the proof of the uniqueness theorem using Bender's method. Whereas in the CN-case, the resulting maximal subgroups "M" are still Frobenius groups, the maximal subgroups that occur in the proof of the odd-order theorem need no longer have this structure, and the analysis of their structure and interplay produces 5 possible types of maximal subgroups, called types I, II, III, IV, V. Type I subgroups are of "Frobenius type", a slight generalization of Frobenius group, and in fact later on in the proof are shown to be Frobenius groups. They have the structure "M""F"⋊"U" where "M""F" is the largest normal nilpotent Hall subgroup, and "U" has a subgroup "U"0 with the same exponent such that "M""F"⋊"U"0 is a Frobenius group with kernel "M""F". Types II, III, IV, V are all 3-step groups with structure "M""F"⋊"U"⋊"W"1, where "M""F"⋊"U" is the derived subgroup of "M". The subdivision into types II, III, IV and V depends on the structure and embedding of the subgroup "U" as follows:
All but two classes of maximal subgroups are of type I, but there may also be two extra classes of maximal subgroups, one of type II, and one of type II, III, IV, or V.
Step 2. Character theory of "G".
If X is an irreducible character of the normalizer "H" of the maximal abelian subgroup "A" of the CA group "G", not containing "A" in its kernel, we can induce X to a character Y of "G", which is not necessarily irreducible. Because of the known structure of "G", it is easy to find the character values of Y on all but the identity element of "G". This implies that if X1 and X2 are two such irreducible characters of "H" and Y1 and Y2 are the corresponding induced characters, then Y1 − Y2 is completely determined, and calculating its norm shows that it is the difference of two irreducible characters of "G" (these are sometimes known as exceptional characters of "G" with respect to "H"). A counting argument shows that each non-trivial irreducible character of "G" arises exactly once as an exceptional character associated to the normalizer of some maximal abelian subgroup of "G". A similar argument (but replacing abelian Hall subgroups by nilpotent Hall subgroups) works in the proof of the CN-theorem. However, in the proof of the odd-order theorem, the arguments for constructing characters of "G" from characters of subgroups are far more delicate, and use the Dade isometry between character rings rather than character induction, since the maximal subgroups have a more complicated structure and are embedded in a less transparent way. The theory of exceptional characters is replaced by the theory of a coherent set of characters to extend the Dade isometry. Roughly speaking, this theory says that the Dade isometry can be extended unless the groups involved have a certain precise structure. described a simplified version the character theory due to Dade, Sibley, and Peterfalvi.
Step 3. The final contradiction.
By step 2, we have a complete and precise description of the character table of the CA group "G". From this, and using the fact that "G" has odd order, sufficient information is available to obtain estimates for |"G"| and arrive at a contradiction to the assumption that "G" is simple. This part of the argument works similarly in the CN-group case.
In the proof of the Feit–Thompson theorem, however, this step is (as usual) vastly more complicated. The character theory only eliminates some of the possible configurations left after step 1. First they show that the maximal subgroups of type I are all Frobenius groups. If all maximal subgroups are type I then an argument similar to the CN case shows that the group "G" cannot be an odd-order minimal simple group, so there are exactly two classes of maximal subgroups of types II, III, IV or V. Most of the rest of the proof now focuses on these two types of maximal subgroup "S" and "T" and the relation between them. More character-theoretic arguments show that they cannot be of types IV or V. The two subgroups have a precise structure: the subgroup "S" is of order "p""q"×"q"×("p""q"–1)/("p"–1) and consists of all automorphisms of the underlying set of the finite field of order "p""q" of the form "x"→"ax""σ"+"b" where "a" has norm 1 and "σ" is an automorphism of the finite field, where "p" and "q" are distinct primes. The maximal subgroup "T" has a similar structure with "p" and "q" reversed. The subgroups "S" and "T" are closely linked. Taking "p">"q", one can show that the cyclic subgroup of "S" of order ("p""q"–1)/("p"–1) is conjugate to a subgroup of the cyclic subgroup of "T" of order ("q""p"–1)/("q"–1). (In particular, the first number divides the second, so if the Feit–Thompson conjecture is true, it would assert that this cannot happen, and this could be used to finish the proof at this point. The conjecture is still unproven, however.)
The conclusion from applying character theory to the group "G" is that "G" has the following structure: there are primes "p">"q" such that ("p""q"–1)/("p"–1) is coprime to "p"–1 and "G" has a subgroup given by the semidirect product "PU" where "P" is the additive group of a finite field of order "p""q" and "U" its elements of norm 1. Moreover "G" has an abelian subgroup "Q" of order prime to "p" containing an element "y" such that "P"0 normalizes "Q" and ("P"0)"y" normalizes "U", where "P"0 is the additive group of the finite field of order "p". (For "p"=2 a similar configuration occurs in the group SL2(2"q"), with "PU" a Borel subgroup of upper triangular matrices and "Q" the subgroup of order 3 generated by $\scriptstyle y=\left(\begin{smallmatrix}0 & 1 \\ 1 & 1\end{smallmatrix}\right)$.) To eliminate this final case, Thompson used some fearsomely complicated manipulations with generators and relations, which were later simplified by , whose argument is reproduced in . The proof examines the set of elements "a" in the finite field of order "p""q" such that "a" and 2–a both have norm 1. One first checks that this set has at least one element other than 1. Then a rather difficult argument using generators and relations in the group "G" shows that the set is closed under taking inverses. If "a" is in the set and not equal to 1 then the polynomial N((1–"a")"x"+1)–1 has degree "q" and has at least "p" distinct roots given by the elements "x" in F"p", using the fact that "x"→1/(2–"x") maps the set to itself, so "p"≤"q", contradicting the assumption "p">"q".
Use of oddness.
The fact that the order of the group "G" is odd is used in several places in the proof, as follows .
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461822
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abstract_algebra
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In the area of modern algebra known as group theory, the Mathieu group "M22" is a sporadic simple group of order
27 · 32 · 5 · 7 · 11 = 443520
≈ 4×105.
History and properties.
"M22" is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M22 is cyclic of order 12, and the outer automorphism group has order 2.
There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature.
incorrectly claimed that the Schur multiplier of M22 has order 3, and in a correction incorrectly claimed that it has order 6. This caused an error in the title of the paper announcing the discovery of the Janko group J4. showed that the Schur multiplier is in fact cyclic of order 12.
calculated the 2-part of all the cohomology of M22.
Representations.
M22 has a 3-transitive permutation representation on 22 points, with point stabilizer the group PSL3(4), sometimes called M21. This action fixes a Steiner system S(3,6,22) with 77 hexads, whose full automorphism group is the automorphism group M22.2 of M22.
M22 has three rank 3 permutation representations: one on the 77 hexads with point stabilizer 24:A6, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A7.
M22 is the point stabilizer of the action of M23 on 23 points, and also the point stabilizer of the rank 3 action of the Higman–Sims group on 100 = 1+22+77 points.
The triple cover 3.M22 has a 6-dimensional faithful representation over the field with 4 elements.
The 6-fold cover of M22 appears in the centralizer 21+12.3.(M22:2) of an involution of the Janko group J4.
Maximal subgroups.
There are no proper subgroups transitive on all 22 points. There are 8 conjugacy classes of maximal subgroups of "M22" as follows:
Stabilizer of W22 block
There are 2 sets, of 15 each, of simple subgroups of order 168. Those of one type have orbits of 1, 7 and 14; the others have orbits of 7, 8, and 7.
Conjugate to preceding type in M22:2.
A 2-point stabilizer in the sextet group
A one-point stabilizer of M11 (point in orbit of 11)
A non-split group extension of form A6.2
Another one-point stabilizer of M11 (point in orbit of 12)
Conjugacy classes.
There are 12 conjugacy classes, though the two classes of elements of order 11 are fused under an outer automorphism.
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1961349
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abstract_algebra
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Group of units of the ring of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to "n" from the set $\{0,1,\dots,n-1\}$ of "n" non-negative integers form a group under multiplication modulo "n", called the multiplicative group of integers modulo "n". Equivalently, the elements of this group can be thought of as the congruence classes, also known as "residues" modulo "n", that are coprime to "n".
Hence another name is the group of primitive residue classes modulo "n".
In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo "n". Here "units" refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to "n".
This quotient group, usually denoted $(\mathbb{Z}/n\mathbb{Z})^\times$, is fundamental in number theory. It is used in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order is given by Euler's totient function: $|(\mathbb{Z}/n\mathbb{Z})^\times|=\varphi(n).$ For prime "n" the group is cyclic, and in general the structure is easy to describe, but no simple general formula for finding generators is known.
Group axioms.
It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo "n" that are coprime to "n" satisfy the axioms for an abelian group.
Indeed, "a" is coprime to "n" if and only if gcd("a", "n") = 1. Integers in the same congruence class "a" ≡ "b" (mod "n") satisfy gcd("a", "n") = gcd("b", "n"), hence one is coprime to "n" if and only if the other is. Thus the notion of congruence classes modulo "n" that are coprime to "n" is well-defined.
Since gcd("a", "n") = 1 and gcd("b", "n") = 1 implies gcd("ab", "n") = 1, the set of classes coprime to "n" is closed under multiplication.
Integer multiplication respects the congruence classes, that is, "a" ≡ "a' " and "b" ≡ "b' " (mod "n") implies "ab" ≡ "a'b' " (mod "n").
This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity.
Finally, given "a", the multiplicative inverse of "a" modulo "n" is an integer "x" satisfying "ax" ≡ 1 (mod "n").
It exists precisely when "a" is coprime to "n", because in that case gcd("a", "n") = 1 and by Bézout's lemma there are integers "x" and "y" satisfying "ax" + "ny" = 1. Notice that the equation "ax" + "ny" = 1 implies that "x" is coprime to "n", so the multiplicative inverse belongs to the group.
Notation.
The set of (congruence classes of) integers modulo "n" with the operations of addition and multiplication is a ring.
It is denoted $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}/(n)$ (the notation refers to taking the quotient of integers modulo the ideal $n\mathbb{Z}$ or $(n)$ consisting of the multiples of "n").
Outside of number theory the simpler notation $\mathbb{Z}_n$ is often used, though it can be confused with the p-adic integers when "n" is a prime number.
The multiplicative group of integers modulo "n", which is the group of units in this ring, may be written as (depending on the author) $(\mathbb{Z}/n\mathbb{Z})^\times,$ $(\mathbb{Z}/n\mathbb{Z})^*,$ $\mathrm{U}(\mathbb{Z}/n\mathbb{Z}),$ $\mathrm{E}(\mathbb{Z}/n\mathbb{Z})$ (for German "Einheit", which translates as "unit"), $\mathbb{Z}_n^*$, or similar notations. This article uses $(\mathbb{Z}/n\mathbb{Z})^\times.$
The notation $\mathrm{C}_n$ refers to the cyclic group of order "n".
It is isomorphic to the group of integers modulo "n" under addition.
Note that $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}_n$ may also refer to the group under addition.
For example, the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^\times$ for a prime "p" is cyclic and hence isomorphic to the additive group $\mathbb{Z}/(p-1)\mathbb{Z}$, but the isomorphism is not obvious.
Structure.
The order of the multiplicative group of integers modulo "n" is the number of integers in $\{0,1,\dots,n-1\}$ coprime to "n".
It is given by Euler's totient function: $| (\mathbb{Z}/n\mathbb{Z})^\times|=\varphi(n)$ (sequence in the OEIS).
For prime "p", $\varphi(p)=p-1$.
Cyclic case.
The group $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if "n" is 1, 2, 4, "p""k" or 2"p""k", where "p" is an odd prime and "k" > 0. For all other values of "n" the group is not cyclic.
This was first proved by Gauss.
This means that for these "n":
$ (\mathbb{Z}/n\mathbb{Z})^\times \cong \mathrm{C}_{\varphi(n)},$ where $\varphi(p^k)=\varphi(2 p^k)=p^k - p^{k-1}.$
By definition, the group is cyclic if and only if it has a generator "g" (a generating set {"g"} of size one), that is, the powers $g^0,g^1,g^2,\dots,$ give all possible residues modulo "n" coprime to "n" (the first $\varphi(n)$ powers $g^0,\dots,g^{\varphi(n)-1}$ give each exactly once).
A generator of $(\mathbb{Z}/n\mathbb{Z})^\times$ is called a primitive root modulo "n".
If there is any generator, then there are $\varphi(\varphi(n))$ of them.
Powers of 2.
Modulo 1 any two integers are congruent, i.e., there is only one congruence class, [0], coprime to 1. Therefore, $(\mathbb{Z}/1\,\mathbb{Z})^\times \cong \mathrm{C}_1$ is the trivial group with φ(1) = 1 element. Because of its trivial nature, the case of congruences modulo 1 is generally ignored and some authors choose not to include the case of "n" = 1 in theorem statements.
Modulo 2 there is only one coprime congruence class, [1], so $(\mathbb{Z}/2\mathbb{Z})^\times \cong \mathrm{C}_1$ is the trivial group.
Modulo 4 there are two coprime congruence classes, [1] and [3], so $(\mathbb{Z}/4\mathbb{Z})^\times \cong \mathrm{C}_2,$ the cyclic group with two elements.
Modulo 8 there are four coprime congruence classes, [1], [3], [5] and [7]. The square of each of these is 1, so $(\mathbb{Z}/8\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_2,$ the Klein four-group.
Modulo 16 there are eight coprime congruence classes [1], [3], [5], [7], [9], [11], [13] and [15]. $\{\pm 1, \pm 7\}\cong \mathrm{C}_2 \times \mathrm{C}_2,$ is the 2-torsion subgroup (i.e., the square of each element is 1), so $(\mathbb{Z}/16\mathbb{Z})^\times$ is not cyclic. The powers of 3, $\{1, 3, 9, 11\}$ are a subgroup of order 4, as are the powers of 5, $\{1, 5, 9, 13\}.$ Thus $(\mathbb{Z}/16\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_4.$
The pattern shown by 8 and 16 holds for higher powers 2"k", "k" > 2: $\{\pm 1, 2^{k-1} \pm 1\}\cong \mathrm{C}_2 \times \mathrm{C}_2,$ is the 2-torsion subgroup (so $(\mathbb{Z}/2^k\mathbb{Z})^\times $ is not cyclic) and the powers of 3 are a cyclic subgroup of order 2"k" − 2, so $(\mathbb{Z}/2^k\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_{2^{k-2}}.$
General composite numbers.
By the fundamental theorem of finite abelian groups, the group $(\mathbb{Z}/n\mathbb{Z})^\times$ is isomorphic to a direct product of cyclic groups of prime power orders.
More specifically, the Chinese remainder theorem says that if $\;\;n=p_1^{k_1}p_2^{k_2}p_3^{k_3}\dots, \;$ then the ring $\mathbb{Z}/n\mathbb{Z}$ is the direct product of the rings corresponding to each of its prime power factors:
$\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/{p_1^{k_1}}\mathbb{Z}\; \times \;\mathbb{Z}/{p_2^{k_2}}\mathbb{Z} \;\times\; \mathbb{Z}/{p_3^{k_3}}\mathbb{Z}\dots\;\;$
Similarly, the group of units $(\mathbb{Z}/n\mathbb{Z})^\times$ is the direct product of the groups corresponding to each of the prime power factors:
$(\mathbb{Z}/n\mathbb{Z})^\times\cong (\mathbb{Z}/{p_1^{k_1}}\mathbb{Z})^\times \times (\mathbb{Z}/{p_2^{k_2}}\mathbb{Z})^\times \times (\mathbb{Z}/{p_3^{k_3}}\mathbb{Z})^\times \dots\;.$
For each odd prime power $p^{k}$ the corresponding factor $(\mathbb{Z}/{p^{k}}\mathbb{Z})^\times$ is the cyclic group of order $\varphi(p^k)=p^k - p^{k-1}$, which may further factor into cyclic groups of prime-power orders.
For powers of 2 the factor $(\mathbb{Z}/{2^{k}}\mathbb{Z})^\times$ is not cyclic unless "k" = 0, 1, 2, but factors into cyclic groups as described above.
The order of the group $\varphi(n)$ is the product of the orders of the cyclic groups in the direct product.
The exponent of the group, that is, the least common multiple of the orders in the cyclic groups, is given by the Carmichael function $\lambda(n)$ (sequence in the OEIS).
In other words, $\lambda(n)$ is the smallest number such that for each "a" coprime to "n", $a^{\lambda(n)} \equiv 1 \pmod n$ holds.
It divides $\varphi(n)$ and is equal to it if and only if the group is cyclic.
Subgroup of false witnesses.
If "n" is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power "n" − 1, are congruent to 1 modulo "n". (Because the residue 1 when raised to any power is congruent to 1 modulo "n", the set of such elements is nonempty.) One could say, because of Fermat's Little Theorem, that such residues are "false positives" or "false witnesses" for the primality of "n". The number 2 is the residue most often used in this basic primality check, hence 341 = 11 × 31 is famous since 2340 is congruent to 1 modulo 341, and 341 is the smallest such composite number (with respect to 2). For 341, the false witnesses subgroup contains 100 residues and so is of index 3 inside the 300 element multiplicative group mod 341.
Examples.
"n" = 9.
The smallest example with a nontrivial subgroup of false witnesses is 9 = 3 × 3. There are 6 residues coprime to 9: 1, 2, 4, 5, 7, 8. Since 8 is congruent to −1 modulo 9, it follows that 88 is congruent to 1 modulo 9. So 1 and 8 are false positives for the "primality" of 9 (since 9 is not actually prime). These are in fact the only ones, so the subgroup {1,8} is the subgroup of false witnesses. The same argument shows that "n" − 1 is a "false witness" for any odd composite "n".
"n" = 91.
For "n" = 91 (= 7 × 13), there are $\varphi(91)=72$ residues coprime to 91, half of them (i.e., 36 of them) are false witnesses of 91, namely 1, 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, and 90, since for these values of "x", "x"90 is congruent to 1 mod 91.
"n" = 561.
"n" = 561 (= 3 × 11 × 17) is a Carmichael number, thus "s"560 is congruent to 1 modulo 561 for any integer "s" coprime to 561. The subgroup of false witnesses is, in this case, not proper; it is the entire group of multiplicative units modulo 561, which consists of 320 residues.
Examples.
This table shows the cyclic decomposition of $(\mathbb{Z}/n\mathbb{Z})^\times$ and a generating set for "n" ≤ 128. The decomposition and generating sets are not unique; for example,
$ \displaystyle \begin{align}(\mathbb{Z}/35\mathbb{Z})^\times & \cong (\mathbb{Z}/5\mathbb{Z})^\times \times (\mathbb{Z}/7\mathbb{Z})^\times \cong \mathrm{C}_4 \times \mathrm{C}_6 \cong \mathrm{C}_4 \times \mathrm{C}_2 \times \mathrm{C}_3 \cong \mathrm{C}_2 \times \mathrm{C}_{12} \cong (\mathbb{Z}/4\mathbb{Z})^\times \times (\mathbb{Z}/13\mathbb{Z})^\times \\ & \cong (\mathbb{Z}/52\mathbb{Z})^\times \end{align} $
(but $\not\cong \mathrm{C}_{24} \cong \mathrm{C}_8 \times \mathrm{C}_3$). The table below lists the shortest decomposition (among those, the lexicographically first is chosen – this guarantees isomorphic groups are listed with the same decompositions). The generating set is also chosen to be as short as possible, and for "n" with primitive root, the smallest primitive root modulo "n" is listed.
For example, take $(\mathbb{Z}/20\mathbb{Z})^\times$. Then $\varphi(20)=8$ means that the order of the group is 8 (i.e., there are 8 numbers less than 20 and coprime to it); $\lambda(20)=4$ means the order of each element divides 4, that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20). The set {3,19} generates the group, which means that every element of $(\mathbb{Z}/20\mathbb{Z})^\times$ is of the form 3"a" × 19"b" (where "a" is 0, 1, 2, or 3, because the element 3 has order 4, and similarly "b" is 0 or 1, because the element 19 has order 2).
Smallest primitive root mod "n" are (0 if no root exists)
0, 1, 2, 3, 2, 5, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 3, 5, 2, 0, 0, 7, 5, 0, 2, 7, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 2, 3, 0, 0, 6, 0, 3, 0, 0, 5, 5, 0, 3, 3, 0, 0, 2, 5, 0, 0, 0, 3, 2, 0, 2, 3, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 5, 5, 0, 0, 0, 0, 3, 0, 2, 7, 2, 0, 0, 3, 0, 0, 3, 0, ... (sequence in the OEIS)
Numbers of the elements in a minimal generating set of mod "n" are
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 2, ... (sequence in the OEIS)
Notes.
References.
The "Disquisitiones Arithmeticae" has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
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Subgroup mapped to itself under every automorphism of the parent group
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group.
Definition.
A subgroup "H" of a group "G" is called a characteristic subgroup if for every automorphism "φ" of "G", one has φ("H") ≤ "H"; then write "H" char "G".
It would be equivalent to require the stronger condition φ("H") = "H" for every automorphism "φ" of "G", because φ−1("H") ≤ "H" implies the reverse inclusion "H" ≤ φ("H").
Basic properties.
Given "H" char "G", every automorphism of "G" induces an automorphism of the quotient group "G/H", which yields a homomorphism Aut("G") → Aut("G"/"H").
If "G" has a unique subgroup "H" of a given index, then "H" is characteristic in "G".
Related concepts.
Normal subgroup.
A subgroup of "H" that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.
∀φ ∈ Inn("G")： φ["H"] ≤ "H"
Since Inn("G") ⊆ Aut("G") and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic. Here are several examples:
{"e", "a", "b", "ab"} . Consider H
{"e", "a"} and consider the automorphism, T("e")
"e", T("a")
"b", T("b")
"a", T("ab")
"ab"; then T("H") is not contained in "H".
Strictly characteristic subgroup.
A "strictly characteristic subgroup", or a "distinguished subgroup", which is invariant under surjective endomorphisms. For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being "strictly characteristic" is equivalent to "characteristic". This is not the case anymore for infinite groups.
Fully characteristic subgroup.
For an even stronger constraint, a "fully characteristic subgroup" (also, "fully invariant subgroup"; cf. invariant subgroup), "H", of a group "G", is a group remaining invariant under every endomorphism of "G"; that is,
∀φ ∈ End("G")： φ["H"] ≤ "H".
Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. The commutator subgroup of a group is always a fully characteristic subgroup.
Every endomorphism of "G" induces an endomorphism of "G/H", which yields a map End("G") → End("G"/"H").
Verbal subgroup.
An even stronger constraint is verbal subgroup, which is the image of a fully invariant subgroup of a free group under a homomorphism. More generally, any verbal subgroup is always fully characteristic. For any reduced free group, and, in particular, for any free group, the converse also holds: every fully characteristic subgroup is verbal.
Transitivity.
The property of being characteristic or fully characteristic is transitive; if "H" is a (fully) characteristic subgroup of "K", and "K" is a (fully) characteristic subgroup of "G", then "H" is a (fully) characteristic subgroup of "G".
"H" char "K" char "G" ⇒ "H" char "G".
Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.
"H" char "K" ⊲ "G" ⇒ "H" ⊲ "G"
Similarly, while being strictly characteristic (distinguished) is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic.
However, unlike normality, if "H" char "G" and "K" is a subgroup of "G" containing "H", then in general "H" is not necessarily characteristic in "K".
"H" char "G", "H" < "K" < "G" ⇏ "H" char "K"
Containments.
Every subgroup that is fully characteristic is certainly strictly characteristic and characteristic; but a characteristic or even strictly characteristic subgroup need not be fully characteristic.
The center of a group is always a strictly characteristic subgroup, but it is not always fully characteristic. For example, the finite group of order 12, Sym(3) × $\mathbb{Z} / 2 \mathbb{Z}$, has a homomorphism taking ("π", "y") to ((1, 2)"y", 0), which takes the center, $1 \times \mathbb{Z} / 2 \mathbb{Z}$, into a subgroup of Sym(3) × 1, which meets the center only in the identity.
The relationship amongst these subgroup properties can be expressed as:
Subgroup ⇐ Normal subgroup ⇐ Characteristic subgroup ⇐ Strictly characteristic subgroup ⇐ Fully characteristic subgroup ⇐ Verbal subgroup
Examples.
Finite example.
Consider the group "G"
S3 × $\mathbb{Z}_2$ (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of "G" is isomorphic to its second factor $\mathbb{Z}_2$. Note that the first factor, S3, contains subgroups isomorphic to $\mathbb{Z}_2$, for instance {e, (12)}; let $f: \mathbb{Z}_2<\rarr \text{S}_3$ be the morphism mapping $\mathbb{Z}_2$ onto the indicated subgroup. Then the composition of the projection of "G" onto its second factor $\mathbb{Z}_2$, followed by "f", followed by the inclusion of S3 into "G" as its first factor, provides an endomorphism of "G" under which the image of the center, $\mathbb{Z}_2$, is not contained in the center, so here the center is not a fully characteristic subgroup of "G".
Cyclic groups.
Every subgroup of a cyclic group is characteristic.
Subgroup functors.
The derived subgroup (or commutator subgroup) of a group is a verbal subgroup. The torsion subgroup of an abelian group is a fully invariant subgroup.
Topological groups.
The identity component of a topological group is always a characteristic subgroup.
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In mathematics, specifically in ring theory, the simple modules over a ring "R" are the (left or right) modules over "R" that are non-zero and have no non-zero proper submodules. Equivalently, a module "M" is simple if and only if every cyclic submodule generated by a non-zero element of "M" equals "M". Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory.
In this article, all modules will be assumed to be right unital modules over a ring "R".
Examples.
Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic groups of prime order.
If "I" is a right ideal of "R", then "I" is simple as a right module if and only if "I" is a minimal non-zero right ideal: If "M" is a non-zero proper submodule of "I", then it is also a right ideal, so "I" is not minimal. Conversely, if "I" is not minimal, then there is a non-zero right ideal "J" properly contained in "I". "J" is a right submodule of "I", so "I" is not simple.
If "I" is a right ideal of "R", then the quotient module "R"/"I" is simple if and only if "I" is a maximal right ideal: If "M" is a non-zero proper submodule of "R"/"I", then the preimage of "M" under the quotient map "R" → "R"/"I" is a right ideal which is not equal to "R" and which properly contains "I". Therefore, "I" is not maximal. Conversely, if "I" is not maximal, then there is a right ideal "J" properly containing "I". The quotient map "R"/"I" → "R"/"J" has a non-zero kernel which is not equal to "R"/"I", and therefore "R"/"I" is not simple.
Every simple "R"-module is isomorphic to a quotient "R"/"m" where "m" is a maximal right ideal of "R". By the above paragraph, any quotient "R"/"m" is a simple module. Conversely, suppose that "M" is a simple "R"-module. Then, for any non-zero element "x" of "M", the cyclic submodule "xR" must equal "M". Fix such an "x". The statement that "xR" = "M" is equivalent to the surjectivity of the homomorphism "R" → "M" that sends "r" to "xr". The kernel of this homomorphism is a right ideal "I" of "R", and a standard theorem states that "M" is isomorphic to "R"/"I". By the above paragraph, we find that "I" is a maximal right ideal. Therefore, "M" is isomorphic to a quotient of "R" by a maximal right ideal.
If "k" is a field and "G" is a group, then a group representation of "G" is a left module over the group ring "k"["G"] (for details, see the main page on this relationship). The simple "k"["G"]-modules are also known as irreducible representations. A major aim of representation theory is to understand the irreducible representations of groups.
Basic properties of simple modules.
The simple modules are precisely the modules of length 1; this is a reformulation of the definition.
Every simple module is indecomposable, but the converse is in general not true.
Every simple module is cyclic, that is it is generated by one element.
Not every module has a simple submodule; consider for instance the Z-module Z in light of the first example above.
Let "M" and "N" be (left or right) modules over the same ring, and let "f" : "M" → "N" be a module homomorphism. If "M" is simple, then "f" is either the zero homomorphism or injective because the kernel of "f" is a submodule of "M". If "N" is simple, then "f" is either the zero homomorphism or surjective because the image of "f" is a submodule of "N". If "M" = "N", then "f" is an endomorphism of "M", and if "M" is simple, then the prior two statements imply that "f" is either the zero homomorphism or an isomorphism. Consequently, the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma.
The converse of Schur's lemma is not true in general. For example, the Z-module Q is not simple, but its endomorphism ring is isomorphic to the field Q.
Simple modules and composition series.
If "M" is a module which has a non-zero proper submodule "N", then there is a short exact sequence
$0 \to N \to M \to M/N \to 0.$
A common approach to proving a fact about "M" is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for "N" and "M"/"N". If "N" has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules
$\cdots \subset M_2 \subset M_1 \subset M.$
In order to prove the fact this way, one needs conditions on this sequence and on the modules "M""i" /"M""i" +&hairsp;1. One particularly useful condition is that the length of the sequence is finite and each quotient module "M""i" /"M""i" +&hairsp;1 is simple. In this case the sequence is called a composition series for "M". In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module. For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull–Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.
The Jordan–Hölder theorem and the Schreier refinement theorem describe the relationships amongst all composition series of a single module. The Grothendieck group ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over semisimple rings, this is no loss as every module is a semisimple module and so a direct sum of simple modules. Ordinary character theory provides better arithmetic control, and uses simple C"G" modules to understand the structure of finite groups "G". Modular representation theory uses Brauer characters to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the Ext functor and describing the module category in various ways including quivers (whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and Auslander–Reiten theory where the associated graph has a vertex for every indecomposable module.
The Jacobson density theorem.
An important advance in the theory of simple modules was the Jacobson density theorem. The Jacobson density theorem states:
Let "U" be a simple right "R"-module and let "D" = End"R"("U"). Let "A" be any "D"-linear operator on "U" and let "X" be a finite "D"-linearly independent subset of "U". Then there exists an element "r" of "R" such that "x"·"A" = "x"·"r" for all "x" in "X".
In particular, any primitive ring may be viewed as (that is, isomorphic to) a ring of "D"-linear operators on some "D"-space.
A consequence of the Jacobson density theorem is Wedderburn's theorem; namely that any right Artinian simple ring is isomorphic to a full matrix ring of "n"-by-"n" matrices over a division ring for some "n". This can also be established as a corollary of the Artin–Wedderburn theorem.
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abstract_algebra
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Mathematical group based upon a finite number of elements
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups.
The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004.
History.
During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known.
During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields.
Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.
Examples.
Permutation groups.
The symmetric group S"n" on a finite set of "n" symbols is the group whose elements are all the permutations of the "n" symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself. Since there are "n"! ("n" factorial) possible permutations of a set of "n" symbols, it follows that the order (the number of elements) of the symmetric group S"n" is "n"!.
Cyclic groups.
A cyclic group Z"n" is a group all of whose elements are powers of a particular element "a" where "a""n" = "a"0 = e, the identity. A typical realization of this group is as the complex roots of unity. Sending "a" to a primitive root of unity gives an isomorphism between the two. This can be done with any finite cyclic group.
Finite abelian groups.
An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). They are named after Niels Henrik Abel.
An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.
Groups of Lie type.
A group of Lie type is a group closely related to the group "G"("k") of rational points of a reductive linear algebraic group "G" with values in the field "k". Finite groups of Lie type give the bulk of nonabelian finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.
Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclic, symmetric and alternating groups, with the projective special linear groups over prime finite fields, PSL(2, "p") being constructed by Évariste Galois in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan's theorem that the projective special linear group PSL(2, "q") is simple for "q" ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL("n", "q") of finite simple groups. Other classical groups were studied by Leonard Dickson in the beginning of 20th century. In the 1950s Claude Chevalley realized that after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for algebraic groups over an arbitrary field "k", leading to construction of what are now called "Chevalley groups". Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups ("Tits simplicity theorem"). Although it was known since 19th century that other finite simple groups exist (for example, Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the sporadic groups, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their "geometry" in the sense of Tits.
The belief has now become a theorem – the classification of finite simple groups. Inspection of the list of finite simple groups shows that groups of Lie type over a finite field include all the finite simple groups other than the cyclic groups, the alternating groups, the Tits group, and the 26 sporadic simple groups.
Main theorems.
Lagrange's theorem.
For any finite group "G", the order (number of elements) of every subgroup "H" of "G" divides the order of "G". The theorem is named after Joseph-Louis Lagrange.
Sylow theorems.
This provides a partial converse to Lagrange's theorem giving information about how many subgroups of a given order are contained in "G".
Cayley's theorem.
Cayley's theorem, named in honour of Arthur Cayley, states that every group "G" is isomorphic to a subgroup of the symmetric group acting on "G". This can be understood as an example of the group action of "G" on the elements of "G".
Burnside's theorem.
Burnside's theorem in group theory states that if "G" is a finite group of order "p""a""q""b", where "p" and "q" are prime numbers, and "a" and "b" are non-negative integers, then "G" is solvable. Hence each
non-Abelian finite simple group has order divisible by at least three distinct primes.
Feit–Thompson theorem.
The Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963)
Classification of finite simple groups.
The classification of finite simple groups is a theorem stating that every finite simple group belongs to one of the following families:
The finite simple groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference with respect to the case of integer factorization is that such "building blocks" do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.
The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.
Number of groups of a given order.
Given a positive integer "n", it is not at all a routine matter to determine how many isomorphism types of groups of order "n" there are. Every group of prime order is cyclic, because Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group.
If "n" is the square of a prime, then there are exactly two possible isomorphism types of group of order "n", both of which are abelian. If "n" is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order "n", and the number grows very rapidly as the power increases.
Depending on the prime factorization of "n", some restrictions may be placed on the structure of groups of order "n", as a consequence, for example, of results such as the Sylow theorems. For example, every group of order "pq" is cyclic when "q" < "p" are primes with "p" − 1 not divisible by "q". For a necessary and sufficient condition, see cyclic number.
If "n" is squarefree, then any group of order "n" is solvable. Burnside's theorem, proved using group characters, states that every group of order "n" is solvable when "n" is divisible by fewer than three distinct primes, i.e. if "n" = "p""a""q""b", where "p" and "q" are prime numbers, and "a" and "b" are non-negative integers. By the Feit–Thompson theorem, which has a long and complicated proof, every group of order "n" is solvable when "n" is odd.
For every positive integer "n", most groups of order "n" are solvable. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the classification of finite simple groups. For any positive integer "n" there are at most two simple groups of order "n", and there are infinitely many positive integers "n" for which there are two non-isomorphic simple groups of order "n".
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abstract_algebra
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In group theory, a dicyclic group (notation Dic"n" or Q4"n", ⟨"n",2,2⟩) is a particular kind of non-abelian group of order 4"n" ("n" > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2"n", giving the name "di-cyclic". In the notation of exact sequences of groups, this extension can be expressed as:
$1 \to C_{2n} \to \mbox{Dic}_n \to C_2 \to 1. \, $
More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group.
Definition.
For each integer "n" > 1, the dicyclic group Dic"n" can be defined as the subgroup of the unit quaternions generated by
$\begin{align}
a & = e^\frac{i\pi}{n} = \cos\frac{\pi}{n} + i\sin\frac{\pi}{n} \\
x & = j
\end{align}$
More abstractly, one can define the dicyclic group Dic"n" as the group with the following presentation
$\operatorname{Dic}_n = \left\langle a, x \mid a^{2n} = 1,\ x^2 = a^n,\ x^{-1}ax = a^{-1}\right\rangle.\,\!$
Some things to note which follow from this definition:
Thus, every element of Dic"n" can be uniquely written as "a""m""x""l", where 0 ≤ "m" < 2"n" and "l" = 0 or 1. The multiplication rules are given by
It follows that Dic"n" has order 4"n".
When "n" = 2, the dicyclic group is isomorphic to the quaternion group "Q". More generally, when "n" is a power of 2, the dicyclic group is isomorphic to the generalized quaternion group.
Properties.
For each "n" > 1, the dicyclic group Dic"n" is a non-abelian group of order 4"n". (For the degenerate case "n" = 1, the group Dic1 is the cyclic group "C"4, which is not considered dicyclic.)
Let "A" = ⟨"a"⟩ be the subgroup of Dic"n" generated by "a". Then "A" is a cyclic group of order 2"n", so [Dic"n":"A"] = 2. As a subgroup of index 2 it is automatically a normal subgroup. The quotient group Dic"n"/"A" is a cyclic group of order 2.
Dic"n" is solvable; note that "A" is normal, and being abelian, is itself solvable.
Binary dihedral group.
The dicyclic group is a binary polyhedral group — it is one of the classes of subgroups of the Pin group Pin−(2), which is a subgroup of the Spin group Spin(3) — and in this context is known as the binary dihedral group.
The connection with the binary cyclic group "C"2"n", the cyclic group "C""n", and the dihedral group Dih"n" of order 2"n" is illustrated in the diagram at right, and parallels the corresponding diagram for the Pin group. Coxeter writes the "binary dihedral group" as ⟨2,2,"n"⟩ and "binary cyclic group" with angle-brackets, ⟨"n"⟩.
There is a superficial resemblance between the dicyclic groups and dihedral groups; both are a sort of "mirroring" of an underlying cyclic group. But the presentation of a dihedral group would have "x"2 = 1, instead of "x"2 = "a""n"; and this yields a different structure. In particular, Dic"n" is not a semidirect product of "A" and ⟨"x"⟩, since "A" ∩ ⟨"x"⟩ is not trivial.
The dicyclic group has a unique involution (i.e. an element of order 2), namely "x"2 = "a""n". Note that this element lies in the center of Dic"n". Indeed, the center consists solely of the identity element and "x"2. If we add the relation "x"2 = 1 to the presentation of Dic"n" one obtains a presentation of the dihedral group Dih"n", so the quotient group Dic"n"/<"x"2> is isomorphic to Dih"n".
There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations. Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism. The answer is just the dihedral symmetry group Dih"n". For this reason the dicyclic group is also known as the binary dihedral group. Note that the dicyclic group does not contain any subgroup isomorphic to Dih"n".
The analogous pre-image construction, using Pin+(2) instead of Pin−(2), yields another dihedral group, Dih2"n", rather than a dicyclic group.
Generalizations.
Let "A" be an abelian group, having a specific element "y" in "A" with order 2. A group "G" is called a generalized dicyclic group, written as Dic("A", "y"), if it is generated by "A" and an additional element "x", and in addition we have that ["G":"A"] = 2, "x"2 = "y", and for all "a" in "A", "x"−1"ax" = "a"−1.
Since for a cyclic group of even order, there is always a unique element of order 2, we can see that dicyclic groups are just a specific type of generalized dicyclic group.
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abstract_algebra
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In mathematics, specifically in group theory, the Prüfer "p"-group or the p"-quasicyclic group or p"∞-group, Z("p"∞), for a prime number "p" is the unique "p"-group in which every element has "p" different "p"-th roots.
The Prüfer "p"-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups.
The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.
Constructions of Z("p"∞).
The Prüfer "p"-group may be identified with the subgroup of the circle group, U(1), consisting of all "p""n"-th roots of unity as "n" ranges over all non-negative integers:
$\mathbf{Z}(p^\infty)=\{\exp(2\pi i m/p^n) \mid 0 \leq m < p^n,\,n\in \mathbf{Z}^+\} = \{z\in\mathbf{C} \mid z^{(p^n)}=1 \text{ for some } n\in \mathbf{Z}^+\}.\;$
The group operation here is the multiplication of complex numbers.
There is a presentation
$\mathbf{Z}(p^\infty) = \langle\, g_1, g_2, g_3, \ldots \mid g_1^p = 1, g_2^p = g_1, g_3^p = g_2, \dots\,\rangle.$
Here, the group operation in Z("p"∞) is written as multiplication.
Alternatively and equivalently, the Prüfer "p"-group may be defined as the Sylow "p"-subgroup of the quotient group Q"/"Z, consisting of those elements whose order is a power of "p":
$\mathbf{Z}(p^\infty) = \mathbf{Z}[1/p]/\mathbf{Z}$
(where Z[1/"p"] denotes the group of all rational numbers whose denominator is a power of "p", using addition of rational numbers as group operation).
For each natural number "n", consider the quotient group Z/"p""n"Z and the embedding Z/"p""n"Z → Z/"p""n"+1Z induced by multiplication by "p". The direct limit of this system is Z("p"∞):
$\mathbf{Z}(p^\infty) = \varinjlim \mathbf{Z}/p^n \mathbf{Z} .$
If we perform the direct limit in the category of topological groups, then we need to impose a topology on each of the $\mathbf{Z}/p^n \mathbf{Z}$, and take the final topology on $\mathbf{Z}(p^\infty)$. If we wish for $\mathbf{Z}(p^\infty)$ to be Hausdorff, we must impose the discrete topology on each of the $\mathbf{Z}/p^n \mathbf{Z}$, resulting in $\mathbf{Z}(p^\infty)$ to have the discrete topology.
We can also write
$\mathbf{Z}(p^\infty)=\mathbf{Q}_p/\mathbf{Z}_p$
where Q"p" denotes the additive group of "p"-adic numbers and Z"p" is the subgroup of "p"-adic integers.
Properties.
The complete list of subgroups of the Prüfer "p"-group Z("p"∞) = Z[1/"p"]/Z is:
$0 \subsetneq \left({1 \over p}\mathbf{Z}\right)/\mathbf{Z} \subsetneq \left({1 \over p^2}\mathbf{Z}\right)/\mathbf{Z} \subsetneq \left({1 \over p^3}\mathbf{Z}\right)/\mathbf{Z} \subsetneq \cdots \subsetneq \mathbf{Z}(p^\infty)$
(Here $\left({1 \over p^n}\mathbf{Z}\right)/\mathbf{Z}$ is a cyclic subgroup of Z("p"∞) with "p""n" elements; it contains precisely those elements of Z("p"∞) whose order divides "p""n" and corresponds to the set of "pn"-th roots of unity.) The Prüfer "p"-groups are the only infinite groups whose subgroups are totally ordered by inclusion. This sequence of inclusions expresses the Prüfer "p"-group as the direct limit of its finite subgroups. As there is no maximal subgroup of a Prüfer "p"-group, it is its own Frattini subgroup.
Given this list of subgroups, it is clear that the Prüfer "p"-groups are indecomposable (cannot be written as a direct sum of proper subgroups). More is true: the Prüfer "p"-groups are subdirectly irreducible. An abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic "p"-group or to a Prüfer group.
The Prüfer "p"-group is the unique infinite "p"-group that is locally cyclic (every finite set of elements generates a cyclic group). As seen above, all proper subgroups of Z("p"∞) are finite. The Prüfer "p"-groups are the only infinite abelian groups with this property.
The Prüfer "p"-groups are divisible. They play an important role in the classification of divisible groups; along with the rational numbers they are the simplest divisible groups. More precisely: an abelian group is divisible if and only if it is the direct sum of a (possibly infinite) number of copies of Q and (possibly infinite) numbers of copies of Z("p"∞) for every prime "p". The (cardinal) numbers of copies of Q and Z("p"∞) that are used in this direct sum determine the divisible group up to isomorphism.
As an abelian group (that is, as a Z-module), Z("p"∞) is Artinian but not Noetherian. It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian "ring" is Noetherian).
The endomorphism ring of Z("p"∞) is isomorphic to the ring of "p"-adic integers Z"p".
In the theory of locally compact topological groups the Prüfer "p"-group (endowed with the discrete topology) is the Pontryagin dual of the compact group of "p"-adic integers, and the group of "p"-adic integers is the Pontryagin dual of the Prüfer "p"-group.
Notes.
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In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element
fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius.
Structure.
Suppose "G" is a Frobenius group consisting of permutations of a set "X". A subgroup "H" of "G" fixing a point of "X" is called a Frobenius complement. The identity element together with all elements not in any conjugate of "H" form a normal subgroup called the Frobenius kernel "K". (This is a theorem due to ; there is still no proof of this theorem that does not use character theory, although see .) The Frobenius group "G" is the semidirect product of "K" and "H":
$G=K\rtimes H$.
Both the Frobenius kernel and the Frobenius complement have very restricted structures. J. G. Thompson (1960) proved that the Frobenius kernel "K" is a nilpotent group. If "H" has even order then "K" is abelian. The Frobenius complement "H" has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called a Z-group, and in particular must be a metacyclic group: this means it is the extension of two cyclic groups. If a Frobenius complement "H" is not solvable then Zassenhaus showed that it
has a normal subgroup of index 1 or 2 that is the product of SL(2,5) and a metacyclic group of order coprime to 30. In particular, if a Frobenius complement coincides with its derived subgroup, then it is isomorphic with SL(2,5). If a Frobenius complement "H" is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points. A finite group is a Frobenius complement if and only if it has a faithful, finite-dimensional representation over a finite field in which non-identity group elements correspond to linear transformations without nonzero fixed points.
The Frobenius kernel "K" is uniquely determined by "G" as it is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus theorem. In particular a finite group "G" is a Frobenius group in at most one way.
Representation theory.
The irreducible complex representations of a Frobenius group "G" can be read off from those of "H" and "K". There are two types of irreducible representations of "G":
Alternative definitions.
There are a number of group theoretical properties which are interesting on their own right, but which happen to be equivalent to the group possessing a permutation representation that makes it a Frobenius group.
This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of CA groups to be extended to the results on CN groups and finally the odd order theorem.
Assuming that $G = K\rtimes H$ is the semidirect product of the normal subgroup "K" and complement "H", then the following restrictions on centralizers are equivalent to "G" being a Frobenius group with Frobenius complement "H":
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Automorphism group of the Klein quartic
In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the smallest nonabelian simple group after the alternating group A5 with 60 elements, isomorphic to PSL(2, 5).
Definition.
The general linear group GL(2, 7) consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. These have nonzero determinant. The subgroup SL(2, 7) consists of all such matrices with unit determinant. Then PSL(2, 7) is defined to be the quotient group
SL(2, 7) / {I, −I}
obtained by identifying I and −I, where "I" is the identity matrix. In this article, we let "G" denote any group isomorphic to PSL(2, 7).
Properties.
"G" = PSL(2, 7) has 168 elements. This can be seen by counting the possible columns; there are 72 − 1 = 48 possibilities for the first column, then 72 − 7 = 42 possibilities for the second column. We must divide by 7 − 1 = 6 to force the determinant equal to one, and then we must divide by 2 when we identify I and −I. The result is (48 × 42) / (6 × 2) = 168.
It is a general result that PSL("n", "q") is simple for "n", "q" ≥ 2 ("q" being some power of a prime number), unless ("n", "q") = (2, 2) or (2, 3). PSL(2, 2) is isomorphic to the symmetric group S3, and PSL(2, 3) is isomorphic to alternating group A4. In fact, PSL(2, 7) is the second smallest nonabelian simple group, after the alternating group A5 = PSL(2, 5) = PSL(2, 4).
The number of conjugacy classes and irreducible representations is 6. The sizes of conjugacy classes are 1, 21, 42, 56, 24, 24. The dimensions of irreducible representations 1, 3, 3, 6, 7, 8.
Character table
$\begin{array}{r|cccccc}
& 1A_{1} & 2A_{21} & 4A_{42} & 3A_{56} & 7A_{24} & 7B_{24} \\ \hline
\chi_1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\chi_2 & 3 & -1 & 1 & 0 & \sigma & \bar \sigma \\
\chi_3 & 3 & -1 & 1 & 0 & \bar \sigma & \sigma \\
\chi_4 & 6 & 2 & 0 & 0 & -1 & -1 \\
\chi_5 & 7 & -1 &-1 & 1 & 0 & 0 \\
\chi_6 & 8 & 0 & 0 & -1 & 1 & 1 \\
\end{array},$
where:
$\sigma = \frac{-1+i\sqrt{7}}{2}.$
The following table describes the conjugacy classes in terms of the order of an element in the class, the size of the class, the minimum polynomial of every representative in GL(3, 2), and the function notation for a representative in PSL(2, 7). Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL(3, 2) and PSL(2, 7) can be switched arbitrarily.
The order of group is 168 = 3 × 7 × 8, this implies existence of Sylow's subgroups of orders 3, 7 and 8. It is easy to describe the first two, they are cyclic, since any group of prime order is cyclic. Any element of conjugacy class 3"A"56 generates Sylow 3-subgroup. Any element from the conjugacy classes 7"A"24, 7"B"24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a dihedral group of order 8. It can be described as centralizer of any element from the conjugacy class 2"A"21. In the GL(3, 2) representation, a Sylow 2-subgroup consists of the upper triangular matrices.
This group and its Sylow 2-subgroup provide a counter-example for various normal p-complement theorems for "p" = 2.
Actions on projective spaces.
"G" = PSL(2, 7) acts via linear fractional transformation on the projective line P1(7) over the field with 7 elements:
$\text{For } \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{PSL}(2, 7) \text{ and } x \in \mathbb{P}^1\!(7),\ \gamma \cdot x = \frac{ax+b}{cx+d} .$
Every orientation-preserving automorphism of P1(7) arises in this way, and so "G" = PSL(2, 7) can be thought of geometrically as a group of symmetries of the projective line P1(7); the full group of possibly orientation-reversing projective linear automorphisms is instead the order 2 extension PGL(2, 7), and the group of collineations of the projective line is the complete symmetric group of the points.
However, PSL(2, 7) is also isomorphic to PSL(3, 2) (= SL(3, 2) = GL(3, 2)), the special (general) linear group of 3×3 matrices over the field with 2 elements. In a similar fashion, "G" = PSL(3, 2) acts on the projective plane P2(2) over the field with 2 elements — also known as the Fano plane:
For $ \gamma = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \in \text{PSL}(3, 2)\ \ $ and $\
\ \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \in \mathbb{P}^2\!(2),\ \ \gamma \ \cdot \ \mathbf{x} = \begin{pmatrix} ax+by+cz \\ dx+ey+fz \\ gx+hy+iz \end{pmatrix}$
Again, every automorphism of P2(2) arises in this way, and so "G" = PSL(3, 2) can be thought of geometrically as the symmetry group of
this projective plane. The Fano plane can be used to describe multiplication of octonions, so "G" acts on the set of octonion multiplication tables.
Symmetries of the Klein quartic.
The Klein quartic is the projective variety over the complex numbers C defined by the quartic polynomial
"x"3"y" + "y"3"z" + "z"3"x" = 0.
It is a compact Riemann surface of genus g = 3, and is the only such surface for which the size of the conformal automorphism group attains the maximum of 84("g"−1). This bound is due to the Hurwitz automorphisms theorem, which holds for all "g">1. Such "Hurwitz surfaces" are rare; the next genus for which any exist is "g" = 7, and the next after that is "g" = 14.
As with all Hurwitz surfaces, the Klein quartic can be given a metric of constant negative curvature and then tiled with regular (hyperbolic) heptagons, as a quotient of the order-3 heptagonal tiling, with the symmetries of the surface as a Riemannian surface or algebraic curve exactly the same as the symmetries of the tiling. For the Klein quartic this yields a tiling by 24 heptagons, and the order of "G" is thus related to the fact that 24 × 7 = 168. Dually, it can be tiled with 56 equilateral triangles, with 24 vertices, each of degree 7, as a quotient of the order-7 triangular tiling.
Klein's quartic arises in many fields of mathematics, including representation theory, homology theory, octonion multiplication, Fermat's Last Theorem, and Stark's theorem on imaginary quadratic number fields of class number 1.
Mathieu group.
PSL(2, 7) is a maximal subgroup of the Mathieu group M21; the groups M21 and M24 can be constructed as extensions of PSL(2, 7). These extensions can be interpreted in terms of the tiling of the Klein quartic, but are not realized by geometric symmetries of the tiling.
Permutation actions.
The group PSL(2, 7) acts on various finite sets:
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In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite "p"-group. They were introduced in where they were used to describe a class of finite "p"-groups whose structure was sufficiently similar to that of finite abelian "p"-groups, the so-called, regular p-groups. The relationship between power and commutator structure forms a central theme in the modern study of "p"-groups, as exemplified in the work on uniformly powerful p-groups.
The word "agemo" is just "omega" spelled backwards, and the agemo subgroup is denoted by an upside-down omega.
Definition.
The omega subgroups are the series of subgroups of a finite p-group, "G", indexed by the natural numbers:
$\Omega_i(G) = \langle \{g : g^{p^i} = 1 \} \rangle. $
The agemo subgroups are the series of subgroups:
$ \mho^i(G) = \langle \{ g^{p^i} : g \in G \} \rangle. $
When "i" = 1 and "p" is odd, then "i" is normally omitted from the definition. When "p" is even, an omitted "i" may mean either "i" = 1 or "i" = 2 depending on local convention. In this article, we use the convention that an omitted "i" always indicates "i" = 1.
Examples.
The dihedral group of order 8, "G", satisfies: ℧("G") = Z("G") = [ "G", "G" ] = Φ("G") = Soc("G") is the unique normal subgroup of order 2, typically realized as the subgroup containing the identity and a 180° rotation. However Ω("G") = "G" is the entire group, since "G" is generated by reflections. This shows that Ω("G") need not be the set of elements of order "p".
The quaternion group of order 8, "H", satisfies Ω("H") = ℧("H") = Z("H") = [ "H", "H" ] = Φ("H") = Soc("H") is the unique subgroup of order 2, normally realized as the subgroup containing only 1 and −1.
The Sylow "p"-subgroup, "P", of the symmetric group on "p"2 points is the wreath product of two cyclic groups of prime order. When "p" = 2, this is just the dihedral group of order 8. It too satisfies Ω("P") = "P". Again ℧("P") = Z("P") = Soc("P") is cyclic of order "p", but [ "P", "P" ] = Φ("G") is elementary abelian of order "p""p"−1.
The semidirect product of a cyclic group of order 4 acting non-trivially on a cyclic group of order 4,
$ K = \langle a,b : a^4 = b^4 = 1, ba=ab^3 \rangle,$
has ℧("K") elementary abelian of order 4, but the set of squares is simply { 1, "aa", "bb" }. Here the element "aabb" of ℧("K") is not a square, showing that ℧ is not simply the set of squares.
Properties.
In this section, let "G" be a finite "p"-group of order |"G"| = "p""n" and exponent exp("G") = "p""k". Then the omega and agemo families satisfy a number of useful properties.
"G" = ℧0("G") ≥ ℧1("G") ≥ ℧2("G") ≥ ... ≥ ℧"k"−2("G") ≥ ℧"k"−1("G") > ℧"k"("G") = 1
"G" = Ω"k"("G") ≥ Ω"k"−1("G") ≥ Ω"k"−2("G") ≥ ... ≥ Ω2("G") ≥ Ω1("G") > Ω0("G") = 1
and the series are loosely intertwined: For all "i" between 1 and "k":
℧"i"("G") ≤ Ω"k"−"i"("G"), but
℧"i"−1("G") is not contained in Ω"k"−"i"("G").
If "H" ≤ "G" is a subgroup of "G" and "N" ⊲ "G" is a normal subgroup of "G", then:
|℧"i"("G")|⋅|Ω"i"("G")| = |"G"|
[℧"i"("G"):℧"i"+1("G")] = [Ω"i"("G"):Ω"i"+1("G")],
where |"H"| is the order of "H" and ["H":"K"] = |"H"|/|"K"| denotes the index of the subgroups "K" ≤ "H".
Applications.
The first application of the omega and agemo subgroups was to draw out the analogy of regular "p"-groups with abelian "p"-groups in .
Groups in which Ω("G") ≤ Z("G") were studied by John G. Thompson and have seen several more recent applications.
The dual notion, groups with ["G","G"] ≤ ℧("G") are called powerful p-groups and were introduced by Avinoam Mann. These groups were critical for the proof of the coclass conjectures which introduced an important way to understand the structure and classification of finite "p"-groups.
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In mathematical group theory, the Thompson replacement theorem is a theorem about the existence of certain abelian subgroups of a "p"-group. The Glauberman replacement theorem is a generalization of it introduced by Glauberman (1968, Theorem 4.1).
Statement.
Suppose that "P" is a finite "p"-group for some prime "p", and let A be the set of abelian subgroups of "P" of maximal order. Suppose that "B" is some abelian subgroup of "P". The Thompson replacement theorem says that if "A" is an element of A that normalizes "B" but is not normalized by "B", then there is another element "A"* of A such that "A"*∩"B" is strictly larger than "A"∩"B", and ["A"*,"A"] normalizes "A".
The Glauberman replacement theorem is similar, except "p" is assumed to be odd and the condition that "B" is abelian is weakened to the condition that ["B","B"] commutes with "B" and with all elements of A. Glauberman says in his paper that he does not know whether the condition that "p" is odd is necessary.
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abstract_algebra
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In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory.
Definitions.
A formation is a topological group "G" together with a topological "G"-module "A" on which "G" acts continuously.
A layer "E"/"F" of a formation is a pair of open subgroups "E", "F" of "G" such that "F" is a finite index subgroup of "E". It is called a normal layer if
"F" is a normal subgroup of "E", and a cyclic layer if in addition the quotient group is cyclic.
If "E" is a subgroup of "G", then "A""E" is defined to be the elements of "A" fixed by "E".
We write
"H""n"("E"/"F")
for the Tate cohomology group
"H""n"("E"/"F", "A""F") whenever "E"/"F" is a normal layer. (Some authors think of "E" and "F" as fixed fields rather than subgroup of "G", so write "F"/"E" instead of "E"/"F".)
In applications, "G" is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure.
A class formation is a formation
such that for every normal layer "E"/"F"
"H"1("E"/"F") is trivial, and
"H"2("E"/"F") is cyclic of order |"E"/"F"|.
In practice, these cyclic groups come provided with canonical generators "u""E"/"F" ∈ "H"2("E"/"F"),
called fundamental classes, that are compatible with each other in the sense that
the restriction (of cohomology classes) of a fundamental class is another fundamental class.
Often the fundamental classes are considered to be part of the structure of a class formation.
A formation that satisfies just the condition "H"1("E"/"F")=1 is sometimes called a field formation.
For example, if "G" is any finite group acting on a field "L" and "A=L×", then this is a field formation by Hilbert's theorem 90.
Examples.
The most important examples of class formations (arranged roughly in order of difficulty) are as follows:
It is easy to verify the class formation property for the finite field case and the archimedean local field case, but the remaining cases are more difficult. Most of the hard work of class field theory consists of proving that these are indeed class formations. This is done in several steps, as described in the sections below.
The first inequality.
The "first inequality" of class field theory states that
|"H"0("E"/"F")| ≥ |"E"/"F"|
for cyclic layers "E"/"F".
It is usually proved using properties of the Herbrand quotient, in the more precise form
|"H"0("E"/"F")| = |"E"/"F"|×|"H"1("E"/"F")|.
It is fairly straightforward to prove, because the Herbrand quotient is easy to work out, as it is multiplicative on short exact sequences, and is 1 for finite modules.
Before about 1950, the first inequality was known as the second inequality, and vice versa.
The second inequality.
The second inequality of class field theory states that
|"H"0("E"/"F")| ≤ |"E"/"F"|
for all normal layers "E"/"F".
For local fields, this inequality follows easily from Hilbert's theorem 90 together with the first inequality and some basic properties of group cohomology.
The second inequality was first proved for global fields by Weber using properties of the L series of number fields, as follows. Suppose that the layer "E"/"F" corresponds to an extension "k"⊂"K" of global fields. By studying the Dedekind zeta function of "K" one shows that the degree 1 primes of "K" have Dirichlet density given by the order of the pole at "s"=1, which is 1 (When "K" is the rationals, this is essentially Euler's proof that there are infinitely many primes using the pole at "s"=1 of the Riemann zeta function.) As each prime in "k" that is a norm is the product of deg("K"/"k")= |"E"/"F"| distinct degree 1 primes of "K", this shows that the set of primes of "k" that are norms has density 1/|"E"/"F"|. On the other hand, by studying Dirichlet L-series of characters of the group "H"0("E"/"F"), one shows that the Dirichlet density of primes of "k" representing the trivial element of this group has density
1/|"H"0("E"/"F")|.
(This part of the proof is a generalization of Dirichlet's proof that there are infinitely many primes in arithmetic progressions.) But a prime represents a trivial element of the group "H"0("E"/"F") if it is equal to a norm modulo principal ideals, so this set is at least as dense as the set of primes that are norms. So
1/|"H"0("E"/"F")| ≥ 1/|"E"/"F"|
which is the second inequality.
In 1940 Chevalley found a purely algebraic proof of the second inequality, but it is longer and harder than Weber's original proof. Before about 1950, the second inequality was known as the first inequality; the name was changed because Chevalley's algebraic proof of it uses the first inequality.
Takagi defined a class field to be one where equality holds in the second inequality. By the Artin isomorphism below, "H"0("E"/"F") is isomorphic to the abelianization of "E"/"F", so equality in the second inequality holds exactly for
abelian extensions, and class fields are the same as abelian extensions.
The first and second inequalities can be combined as follows. For cyclic layers, the two inequalities together prove that
"H"1("E"/"F")|"E"/"F"| = "H"0("E"/"F") ≤ |"E"/"F"|
so
"H"0("E"/"F") = |"E"/"F"|
and
"H"1("E"/"F") = 1.
Now a basic theorem about cohomology groups shows that since "H"1("E"/"F") = 1 for all cyclic layers, we have
"H"1("E"/"F") = 1
for all normal layers (so in particular the formation is a field formation).
This proof that "H"1("E"/"F") is always trivial is rather roundabout; no "direct" proof of it (whatever this means) for global fields is known. (For local fields the vanishing of "H"1("E"/"F") is just Hilbert's theorem 90.)
For cyclic group, "H"0 is the same as "H"2, so "H"2("E"/"F") = |"E"/"F"| for all cyclic layers.
Another theorem of group cohomology shows that since "H"1("E"/"F") = 1
for all normal layers and "H"2("E"/"F") ≤ |"E"/"F"| for all cyclic layers, we have
"H"2("E"/"F")≤ |"E"/"F"|
for all normal layers. (In fact, equality holds for all normal layers, but this takes more work; see the next section.)
The Brauer group.
The Brauer groups "H"2("E"/*) of a class formation are defined to be the direct limit of the groups "H"2("E"/"F") as "F" runs over all open subgroups of "E". An easy consequence of the vanishing of "H"1 for all layers is that the groups "H"2("E"/"F") are all subgroups of the Brauer group. In local class field theory the Brauer groups are the same as Brauer groups of fields, but in global class field theory the Brauer group of the formation is not the Brauer group of the corresponding global field (though they are related).
The next step is to prove that "H"2("E"/"F") is cyclic of order exactly |"E"/"F"|; the previous section shows that it has at most this order, so it is sufficient to find some element of order |"E"/"F"| in "H"2("E"/"F").
The proof for arbitrary extensions uses a homomorphism from the group "G" onto the profinite completion of the integers with kernel "G"∞, or in other words a compatible sequence of homomorphisms of "G" onto the cyclic groups of order "n" for all "n", with kernels "G""n". These homomorphisms are constructed using cyclic cyclotomic extensions of fields; for finite fields they are given by the algebraic closure, for non-archimedean local fields they are given by the maximal unramified extensions, and for global fields they are slightly more complicated. As these extensions are given explicitly one can check that they have the property that H2("G"/"G""n") is cyclic of order "n", with a canonical generator. It follows from this that for any layer "E", the group H2("E"/"E"∩"G"∞) is canonically isomorphic to Q/Z. This idea of using roots of unity was introduced by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem.
For general layers "E","F" there is an exact sequence
$0\rightarrow H^2(E/F)\cap H^2(E/E\cap G_\infty) \rightarrow H^2(E/E\cap G_\infty)\rightarrow H^2(F/F\cap G_\infty)$
The last two groups in this sequence can both be identified with Q/Z and the map between them is then multiplication by |"E"/"F"|. So the first group is canonically isomorphic to Z/"nZ. As "H"2("E"/"F") has order at most Z/"nZ is must be equal to Z/"n"Z (and in particular is contained in the middle group)).
This shows that the second cohomology group "H"2("E"/"F") of any layer is cyclic of order |"E"/"F"|, which completes the verification of the axioms of a class formation.
With a little more care in the proofs, we get a canonical generator of "H"2("E"/"F"), called the fundamental class.
It follows from this that the Brauer group "H"2("E"/*) is (canonically) isomorphic to the group Q/Z, except in the case of the archimedean local fields R and C when it has order 2 or 1.
Tate's theorem and the Artin map.
Tate's theorem in group cohomology is as follows. Suppose that "A" is a module over a finite group "G" and "a" is an element of "H"2("G","A"), such that for every subgroup "E" of "G"
Then cup product with "a" is an isomorphism
If we apply the case "n"=−2 of Tate's theorem to a class formation, we find that there is an isomorphism
for any normal layer "E"/"F". The group "H""−2"("E"/"F",Z) is just the abelianization of "E"/"F", and the group "H"0("E"/"F","A""F") is "A""E" modulo the group of norms of "A""F". In other words, we have an explicit description of the abelianization of the Galois group "E"/"F" in terms of "A""E".
Taking the inverse of this isomorphism gives a homomorphism
"A""E" → abelianization of "E"/"F",
and taking the limit over all open subgroups "F" gives a homomorphism
"A""E" → abelianization of "E",
called the Artin map. The Artin map is not necessarily surjective, but has dense image. By the existence theorem below its kernel is the connected component of "A""E" (for class field theory), which is trivial for class field theory of non-archimedean local fields and for function fields, but is non-trivial for archimedean local fields and number fields.
The Takagi existence theorem.
The main remaining theorem of class field theory is the Takagi existence theorem, which states that every
finite index closed subgroup of the idele class group is the group of norms corresponding to some abelian extension.
The classical way to prove this is to construct some extensions with small groups of norms, by first adding in many roots of unity, and then taking Kummer extensions and Artin–Schreier extensions. These extensions may be non-abelian (though they are extensions of abelian groups by abelian groups); however, this does not really matter, as the norm group of a non-abelian Galois extension is the same as that of its maximal abelian extension (this can be shown using what we already know about class fields). This gives enough (abelian) extensions to show that there is an abelian extension corresponding to any finite index subgroup of the idele class group.
A consequence is that the kernel of the Artin map is the connected component of the identity of the idele class group, so that the abelianization of the Galois group of "F" is the profinite completion of the idele class group.
For local class field theory, it is also possible to construct abelian extensions more explicitly using Lubin–Tate formal group laws. For global fields, the abelian extensions can be constructed explicitly in some cases: for example, the abelian extensions of the rationals can be constructed using roots of unity, and the abelian extensions of quadratic imaginary fields can be constructed using elliptic functions, but finding an analog of this for arbitrary global fields is an unsolved problem.
"This is not a Weyl group and has no connection with the Weil–Châtelet group or the Mordell–Weil group"
Weil group.
The Weil group of a class formation with fundamental classes "u""E"/"F" ∈ "H"2("E"/"F", "A""F") is a kind of modified Galois group, introduced by and used in various formulations of class field theory, and in particular in the Langlands program.
If "E"/"F" is a normal layer, then the Weil group "U" of "E"/"F" is the extension
1 → "A""F" → "U" → "E"/"F" → 1
corresponding to the fundamental class "u""E"/"F" in "H"2("E"/"F", "A""F"). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers
"G"/"F", for "F" an open subgroup of "G".
The reciprocity map of the class formation ("G", "A") induces an isomorphism from "AG" to the abelianization of the Weil group.
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In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written for The Santa Cruz Conference on Finite Groups in 1979, with a footnote that Michael O'Nan had independently proved the same result. Michael Aschbacher and Scott later gave a corrected version of the statement of the theorem.
The theorem states that a maximal subgroup of the symmetric group Sym(Ω), where |Ω| = "n", is one of the following:
* AGL("d","p")
*"Sl "wr" Sk", the stabilizer of the product structure Ω = Δ"k"
*a group of diagonal type
*an almost simple group
In a survey paper written for the Bulletin of the London Mathematical Society,
Peter J. Cameron seems to have been the first to recognize that the real power in the O'Nan–Scott theorem is in the ability to split the finite primitive groups into various types.
A complete version of the theorem with a self-contained proof was given by M.W. Liebeck, Cheryl Praeger and Jan Saxl. The theorem is now a standard part of textbooks on permutation groups.
O'Nan–Scott types.
The eight O'Nan–Scott types of finite primitive permutation groups are as follows:
HA (holomorph of an abelian group): These are the primitive groups which are subgroups of the affine general linear group AGL("d","p"), for some prime "p" and positive integer "d" ≥ 1. For such a group "G" to be primitive, it must contain the subgroup of all translations, and the stabilizer G0 in "G" of the zero vector must be an irreducible subgroup of GL("d,p"). Primitive groups of type HA are characterized by having a unique minimal normal subgroup which is elementary abelian and acts regularly.
HS (holomorph of a simple group): Let "T" be a finite nonabelian simple group. Then "M" = "T"×"T" acts on Ω = "T" by "t"("t"1,"t"2) = "t"1−1"tt"2. Now "M" has two minimal normal subgroups "N"1, "N"2, each isomorphic to "T" and each acts regularly on Ω, one by right multiplication and one by left multiplication. The action of "M" is primitive and if we take "α" = 1"T" we have "M""α" = {("t","t")|"t" ∈ "T"}, which includes Inn("T") on Ω. In fact any automorphism of "T" will act on Ω. A primitive group of type HS is then any group "G" such that "M" ≅ "T".Inn("T") ≤ "G" ≤ "T".Aut("T"). All such groups have "N"1 and "N"2 as minimal normal subgroups.
HC (holomorph of a compound group): Let "T" be a nonabelian simple group and let "N"1 ≅ "N"2 ≅ "T""k" for some integer "k" ≥ 2. Let Ω = "T""k". Then "M" = "N"1 × "N"2 acts transitively on Ω via "x"("n"1,"n"2) = "n"1−1"xn"2 for all "x" ∈ Ω, "n"1 ∈ "N"1, "n"2 ∈ "N"2. As in the HS case, we have "M" ≅ "T""k".Inn("T""k") and any automorphism of "T""k" also acts on Ω. A primitive group of type HC is a group "G" such that "M" ≤ "G" ≤ "T""k".Aut("T""k")and "G" induces a subgroup of Aut("T""k") = Aut("T")wr"S""k" which acts transitively on the set of "k" simple direct factors of "T""k". Any such "G" has two minimal normal subgroups, each isomorphic to "T""k" and regular.
A group of type HC preserves a product structure Ω = Δ"k" where Δ = "T" and "G"≤ "H"wr"S""k" where "H" is a primitive group of type HS on Δ.
TW (twisted wreath): Here "G" has a unique minimal normal subgroup "N" and "N" ≅ "T""k" for some finite nonabelian simple group "T" and "N" acts regularly on Ω. Such groups can be constructed as twisted wreath products and hence the label TW. The conditions required to get primitivity imply that "k"≥ 6 so the smallest degree of such a primitive group is 606 .
AS (almost simple): Here "G" is a group lying between "T" and Aut("T" ), that is, "G" is an almost simple group and so the name. We are not told anything about what the action is, other than that it is primitive. Analysis of this type requires knowing about the possible primitive actions of almost simple groups, which is equivalent to knowing the maximal subgroups of almost simple groups.
SD (simple diagonal): Let "N" = "T""k" for some nonabelian simple group "T" and integer "k" ≥ 2 and let "H" = {("t...,t")| "t" ∈ "T"} ≤ "N". Then "N" acts on the set Ω of right cosets of "H" in "N" by right multiplication. We can take {("t"1...,"t""k"−1, 1)| "t""i" ∈ "T"}to be a set of coset representatives for "H" in "N" and so we can identify Ω with "T""k"−1. Now ("s"1...,"s""k") ∈ "N" takes the coset with representative ("t"1...,"t""k"−1, 1) to the coset "H"("t"1"s"1...,"t""k"−1"s""k"−1, "s""k") = "H"("s""k"−1"t""k""s"1...,"s""k"−1"t""k"−1"s""k"−1, 1). The group "S""k" induces automorphisms of "N" by permuting the entries and fixes the subgroup "H" and so acts on the set Ω. Also, note that "H" acts on Ω by inducing Inn("T") and in fact any automorphism σ of "T" acts on Ω by taking the coset with representative ("t"1...,"t""k"−1, 1)to the coset with representative ("t"1σ...,"t""k"−1σ, 1). Thus we get a group "W" = "N".(Out("T") × "Sk") ≤ Sym(Ω). A primitive group of type SD is a group "G" ≤ "W" such that "N" ◅ "G" and "G" induces a primitive subgroup of "S""k" on the "k" simple direct factors of "N".
CD (compound diagonal): Here Ω = Δ"k" and "G" ≤ "H"wr"Sk" where "H" is a primitive group of type SD on Δ with minimal normal subgroup "Tl". Moreover, "N" = "T""kl" is a minimal normal subgroup of "G" and "G" induces a transitive subgroup of "S""k".
PA (product action): Here Ω = Δ"k" and "G" ≤ "H"wr"S""k" where "H" is a primitive almost simple group on Δ with socle "T". Thus "G" has a product action on Ω. Moreover, "N" = "T""k" ◅ "G" and "G" induces a transitive subgroup of "Sk" in its action on the "k" simple direct factors of "N."
Some authors use different divisions of the types. The most common is to include types HS and SD together as a “diagonal type” and types HC, CD and PA together as a “product action type." Praeger later generalized the O’Nan–Scott Theorem to quasiprimitive groups (groups with faithful actions such that the restriction to every nontrivial normal subgroup is transitive).
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Mathematics group theory concept
In mathematics, specifically group theory, the index of a subgroup "H" in a group "G" is the
number of left cosets of "H" in "G", or equivalently, the number of right cosets of "H" in "G".
The index is denoted $|G:H|$ or $[G:H]$ or $(G:H)$.
Because "G" is the disjoint union of the left cosets and because each left coset has the same size as "H", the index is related to the orders of the two groups by the formula
$|G| = |G:H| |H|$
(interpret the quantities as cardinal numbers if some of them are infinite).
Thus the index $|G:H|$ measures the "relative sizes" of "G" and "H".
For example, let $G = \Z$ be the group of integers under addition, and let $H = 2\Z$ be the subgroup consisting of the even integers. Then $2\Z$ has two cosets in $\Z$, namely the set of even integers and the set of odd integers, so the index $|\Z:2\Z|$ is 2. More generally, $|\Z:n\Z| = n$ for any positive integer "n".
When "G" is finite, the formula may be written as $|G:H| = |G|/|H|$, and it implies
Lagrange's theorem that $|H|$ divides $|G|$.
When "G" is infinite, $|G:H|$ is a nonzero cardinal number that may be finite or infinite.
For example, $|\Z:2\Z| = 2$, but $|\R:\Z|$ is infinite.
If "N" is a normal subgroup of "G", then $|G:N|$ is equal to the order of the quotient group $G/N$, since the underlying set of $G/N$ is the set of cosets of "N" in "G".
$|G:K| = |G:H|\,|H:K|.$
$|G:H\cap K| \le |G : H|\,|G : K|,$
with equality if $HK=G$. (If $|G:H\cap K|$ is finite, then equality holds if and only if $HK=G$.)
$|H:H\cap K| \le |G:K|,$
with equality if $HK=G$. (If $|H:H\cap K|$ is finite, then equality holds if and only if $HK=G$.)
$|G:\operatorname{ker}\;\varphi|=|\operatorname{im}\;\varphi|.$
$|Gx| = |G:G_x|.\!$
This is known as the orbit-stabilizer theorem.
$|G:\operatorname{Core}(H)| \le |G:H|!$
where ! denotes the factorial function; this is discussed further below.
* As a corollary, if the index of "H" in "G" is 2, or for a finite group the lowest prime "p" that divides the order of "G," then "H" is normal, as the index of its core must also be "p," and thus "H" equals its core, i.e., it is normal.
* Note that a subgroup of lowest prime index may not exist, such as in any simple group of non-prime order, or more generally any perfect group.
$\{(x,y) \mid x\text{ is even}\},\quad \{(x,y) \mid y\text{ is even}\},\quad\text{and}\quad
Examples.
\{(x,y) \mid x+y\text{ is even}\}$.
Infinite index.
If "H" has an infinite number of cosets in "G", then the index of "H" in "G" is said to be infinite. In this case, the index $|G:H|$ is actually a cardinal number. For example, the index of "H" in "G" may be countable or uncountable, depending on whether "H" has a countable number of cosets in "G". Note that the index of "H" is at most the order of "G," which is realized for the trivial subgroup, or in fact any subgroup "H" of infinite cardinality less than that of "G."
Finite index.
A subgroup "H" of finite index in a group "G" (finite or infinite) always contains a normal subgroup "N" (of "G"), also of finite index. In fact, if "H" has index "n", then the index of "N" will be some divisor of "n"! and a multiple of "n"; indeed, "N" can be taken to be the kernel of the natural homomorphism from "G" to the permutation group of the left (or right) cosets of "H".
Let us explain this in more detail, using right cosets:
The elements of "G" that leave all cosets the same form a group.
Let us call this group "A". Let "B" be the set of elements of "G" which perform a given permutation on the cosets of "H". Then "B" is a right coset of "A".
What we have said so far applies whether the index of "H" is finite or infinte. Now assume that it is the finite number "n". Since the number of possible permutations of cosets is finite, namely "n"!, then there can only be a finite number of sets like "B". (If "G" is infinite, then all such sets are therefore infinite.) The set of these sets forms a group isomorphic to a subset of the group of permutations, so the number of these sets must divide "n"!. Furthermore, it must be a multiple of "n" because each coset of "H" contains the same number of cosets of "A". Finally, if for some "c" ∈ "G" and "a" ∈ "A" we have "ca = xc", then for any "d" ∈ "G dca = dxc", but also "dca = hdc" for some "h" ∈ "H" (by the definition of "A"), so "hd = dx". Since this is true for any "d", "x" must be a member of A, so "ca = xc" implies that "cac−1" ∈ "A" and therefore "A" is a normal subgroup.
The index of the normal subgroup not only has to be a divisor of "n"!, but must satisfy other criteria as well. Since the normal subgroup is a subgroup of "H", its index in "G" must be "n" times its index inside "H". Its index in "G" must also correspond to a subgroup of the symmetric group S"n", the group of permutations of "n" objects. So for example if "n" is 5, the index cannot be 15 even though this divides 5!, because there is no subgroup of order 15 in S5.
In the case of "n" = 2 this gives the rather obvious result that a subgroup "H" of index 2 is a normal subgroup, because the normal subgroup of "H" must have index 2 in "G" and therefore be identical to "H". (We can arrive at this fact also by noting that all the elements of "G" that are not in "H" constitute the right coset of "H" and also the left coset, so the two are identical.) More generally, a subgroup of index "p" where "p" is the smallest prime factor of the order of "G" (if "G" is finite) is necessarily normal, as the index of "N" divides "p"! and thus must equal "p," having no other prime factors. For example, the subgroup "Z"7 of the non-abelian group of order 21 is normal (see List of small non-abelian groups and Frobenius group#Examples).
An alternative proof of the result that a subgroup of index lowest prime "p" is normal, and other properties of subgroups of prime index are given in .
Examples.
The group O of chiral octahedral symmetry has 24 elements. It has a dihedral D4 subgroup (in fact it has three such) of order 8, and thus of index 3 in O, which we shall call "H". This dihedral group has a 4-member D2 subgroup, which we may call "A". Multiplying on the right any element of a right coset of "H" by an element of "A" gives a member of the same coset of "H" ("Hca = Hc"). "A" is normal in O. There are six cosets of "A", corresponding to the six elements of the symmetric group S3. All elements from any particular coset of "A" perform the same permutation of the cosets of "H".
On the other hand, the group Th of pyritohedral symmetry also has 24 members and a subgroup of index 3 (this time it is a D2h prismatic symmetry group, see point groups in three dimensions), but in this case the whole subgroup is a normal subgroup. All members of a particular coset carry out the same permutation of these cosets, but in this case they represent only the 3-element alternating group in the 6-member S3 symmetric group.
Normal subgroups of prime power index.
Normal subgroups of prime power index are kernels of surjective maps to "p"-groups and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theorem.
There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:
As these are weaker conditions on the groups "K," one obtains the containments
$\mathbf{E}^p(G) \supseteq \mathbf{A}^p(G) \supseteq \mathbf{O}^p(G).$
These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.
Geometric structure.
An elementary observation is that one cannot have exactly 2 subgroups of index 2, as the complement of their symmetric difference yields a third. This is a simple corollary of the above discussion (namely the projectivization of the vector space structure of the elementary abelian group
$G/\mathbf{E}^p(G) \cong (\mathbf{Z}/p)^k$,
and further, "G" does not act on this geometry, nor does it reflect any of the non-abelian structure (in both cases because the quotient is abelian).
However, it is an elementary result, which can be seen concretely as follows: the set of normal subgroups of a given index "p" form a projective space, namely the projective space
$\mathbf{P}(\operatorname{Hom}(G,\mathbf{Z}/p)).$
In detail, the space of homomorphisms from "G" to the (cyclic) group of order "p," $\operatorname{Hom}(G,\mathbf{Z}/p),$ is a vector space over the finite field $\mathbf{F}_p = \mathbf{Z}/p.$ A non-trivial such map has as kernel a normal subgroup of index "p," and multiplying the map by an element of $(\mathbf{Z}/p)^\times$ (a non-zero number mod "p") does not change the kernel; thus one obtains a map from
$\mathbf{P}(\operatorname{Hom}(G,\mathbf{Z}/p)) := (\operatorname{Hom}(G,\mathbf{Z}/p))\setminus\{0\})/(\mathbf{Z}/p)^\times$
to normal index "p" subgroups. Conversely, a normal subgroup of index "p" determines a non-trivial map to $\mathbf{Z}/p$ up to a choice of "which coset maps to $1 \in \mathbf{Z}/p,$ which shows that this map is a bijection.
As a consequence, the number of normal subgroups of index "p" is
$(p^{k+1}-1)/(p-1)=1+p+\cdots+p^k$
for some "k;" $k=-1$ corresponds to no normal subgroups of index "p". Further, given two distinct normal subgroups of index "p," one obtains a projective line consisting of $p+1$ such subgroups.
For $p=2,$ the symmetric difference of two distinct index 2 subgroups (which are necessarily normal) gives the third point on the projective line containing these subgroups, and a group must contain $0,1,3,7,15,\ldots$ index 2 subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.
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In mathematics, the height of an element "g" of an abelian group "A" is an invariant that captures its divisibility properties: it is the largest natural number "N" such that the equation "Nx" = "g" has a solution "x" ∈ "A", or the symbol ∞ if there is no such "N". The "p"-height considers only divisibility properties by the powers of a fixed prime number "p". The notion of height admits a refinement so that the "p"-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants.
Definition of height.
Let "A" be an abelian group and "g" an element of "A". The "p"-height of "g" in "A", denoted "h""p"("g"), is the largest natural number "n" such that the equation "p""n""x" = "g" has a solution in "x" ∈ "A", or the symbol ∞ if a solution exists for all "n". Thus "h""p"("g") = "n" if and only if "g" ∈ "p""n""A" and "g" ∉ "p""n"+1"A".
This allows one to refine the notion of height.
For any ordinal "α", there is a subgroup "p""α""A" of "A" which is the image of the multiplication map by "p" iterated "α" times, defined using
transfinite induction:
The subgroups "p""α""A" form a decreasing filtration of the group "A", and their intersection is the subgroup of the "p"-divisible elements of "A", whose elements are assigned height ∞. The modified "p"-height "h""p"∗("g") = "α" if "g" ∈ "p""α""A", but "g" ∉ "p""α"+1"A". The construction of "p""α""A" is functorial in "A"; in particular, subquotients of the filtration are isomorphism invariants of "A".
Ulm subgroups.
Let "p" be a fixed prime number. The (first) Ulm subgroup of an abelian group "A", denoted "U"("A") or "A"1, is "p""ω""A" = ∩"n" "p""n""A", where "ω" is the smallest infinite ordinal. It consists of all elements of "A" of infinite height. The family {"U""σ"("A")} of Ulm subgroups indexed by ordinals "σ" is defined by transfinite induction:
Equivalently, "U""σ"("A") = "p""ωσ""A", where "ωσ" is the product of ordinals "ω" and "σ".
Ulm subgroups form a decreasing filtration of "A" whose quotients "U""σ"("A") = "U""σ"("A")/"U""σ"+1("A") are called the Ulm factors of "A". This filtration stabilizes and the smallest ordinal "τ" such that "U""τ"("A") = "U""τ"+1("A") is the Ulm length of "A". The smallest Ulm subgroup "U""τ"("A"), also denoted "U"∞("A") and "p"∞A, is the largest "p"-divisible subgroup of "A"; if "A" is a "p"-group, then "U"∞("A") is divisible, and as such it is a direct summand of "A".
For every Ulm factor "U""σ"("A") the "p"-heights of its elements are finite and they are unbounded for every Ulm factor except possibly the last one, namely "U""τ"−1("A") when the Ulm length "τ" is a successor ordinal.
Ulm's theorem.
The second Prüfer theorem provides a straightforward extension of the fundamental theorem of finitely generated abelian groups to countable abelian "p"-groups without elements of infinite height: each such group is isomorphic to a direct sum of cyclic groups whose orders are powers of "p". Moreover, the cardinality of the set of summands of order "p""n" is uniquely determined by the group and each sequence of at most countable cardinalities is realized. Helmut Ulm (1933) found an extension of this classification theory to general countable "p"-groups: their isomorphism class is determined by the isomorphism classes of the Ulm factors and the "p"-divisible part.
Ulm's theorem. "Let" "A" "and" "B" "be countable abelian" "p"-"groups such that for every ordinal" "σ" "their Ulm factors are isomorphic", "U""σ"("A") ≅ "U""σ"("B") "and the" "p"-"divisible parts of" "A" "and" "B" "are isomorphic", "U"∞("A") ≅ "U"∞("B"). "Then" "A" "and" "B" "are isomorphic."
There is a complement to this theorem, first stated by Leo Zippin (1935) and proved in Kurosh (1960), which addresses the existence of an abelian "p"-group with given Ulm factors.
"Let" "τ" "be an ordinal and" {"A""σ"} "be a family of countable abelian" "p"-"groups indexed by the ordinals" "σ" < "τ" "such that the" "p"-"heights of elements of each" "A""σ" "are finite and, except possibly for the last one, are unbounded. Then there exists a reduced abelian" "p"-"group" "A" "of Ulm length" "τ" "whose Ulm factors are isomorphic to these" "p"-"groups", "U""σ"("A") ≅ "A""σ".
Ulm's original proof was based on an extension of the theory of elementary divisors to infinite matrices.
Alternative formulation.
George Mackey and Irving Kaplansky generalized Ulm's theorem to certain modules over a complete discrete valuation ring. They introduced invariants of abelian groups that lead to a direct statement of the classification of countable periodic abelian groups: given an abelian group "A", a prime "p", and an ordinal "α", the corresponding "α"th Ulm invariant is the dimension of the quotient
"p""α""A"["p"]/"p""α"+1"A"["p"],
where "B"["p"] denotes the "p"-torsion of an abelian group "B", i.e. the subgroup of elements of order "p", viewed as a vector space over the finite field with "p" elements.
"A countable periodic reduced abelian group is determined uniquely up to isomorphism by its Ulm invariants for all prime numbers "p" and countable ordinals "α"."
Their simplified proof of Ulm's theorem served as a model for many further generalizations to other classes of abelian groups and modules.
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In mathematics, Clifford theory, introduced by , describes the relation between representations of a group and those of a normal subgroup.
Alfred H. Clifford.
Alfred H. Clifford proved the following result on the restriction of finite-dimensional irreducible representations from a group "G" to a normal subgroup "N" of finite index:
Clifford's theorem.
Theorem. Let π: "G" → GL("n","K") be an irreducible representation with "K" a field. Then the restriction of π to "N" breaks up into a direct sum of irreducible representations of "N" of equal dimensions. These irreducible representations of "N" lie in one orbit for the action of "G" by conjugation on the equivalence classes of irreducible representations of "N". In particular the number of pairwise nonisomorphic summands is no greater than the index of "N" in "G".
Clifford's theorem yields information about the restriction of a complex irreducible character of a finite group "G" to a normal subgroup "N." If μ is a complex character of "N", then for a fixed element "g" of "G", another character, μ(g), of "N" may be constructed by setting
$\mu^{(g)}(n) = \mu(gng^{-1})$
for all "n" in "N". The character μ(g) is irreducible if and only if μ is. Clifford's theorem states that if χ is a complex irreducible character of "G," and μ is an irreducible character of "N" with
$\langle \chi_N,\mu \rangle \neq 0,$ then
$\chi_N = e\left(\sum_{i=1}^{t} \mu^{(g_i)}\right), $
where "e" and "t" are positive integers, and each "gi" is an element of "G." The integers "e" and "t" both divide the index ["G":"N"]. The integer "t" is the index of a subgroup of "G", containing "N", known as the inertial subgroup of μ. This is
$ \{ g \in G: \mu^{(g)} = \mu \}$
and is often denoted by
$I_G(\mu).$
The elements "gi" may be taken to be representatives of all the right cosets of the subgroup "IG"(μ) in "G".
In fact, the integer "e" divides the index
$[I_G(\mu):N],$
though the proof of this fact requires some use of Schur's theory of projective representations.
Proof of Clifford's theorem.
The proof of Clifford's theorem is best explained in terms of modules (and the module-theoretic version works for irreducible modular representations). Let "K" be a field, "V" be an irreducible "K"["G"]-module, "VN" be its restriction to "N" and "U" be an irreducible "K"[N]-submodule of "VN". For each "g" in "G", "U"."g" is an irreducible "K"["N"]-submodule of "VN", and $\sum_{g \in G} U.g $ is an "K"["G"]-submodule of "V", so must be all of "V" by irreducibility. Now "VN" is expressed as a sum of irreducible submodules, and this expression may be refined to a direct sum. The proof of the character-theoretic statement of the theorem may now be completed in the case "K" = C. Let χ be the character of "G" afforded by "V" and μ be the character of "N" afforded by "U". For each "g" in "G", the C["N"]-submodule "U"."g" affords the character μ(g) and $\langle \chi_N,\mu^{(g)}\rangle = \langle \chi_N^{(g)},\mu^{(g)}\rangle = \langle \chi_N,\mu \rangle $. The respective equalities follow because χ is a class-function of "G" and "N" is a normal subgroup. The integer "e" appearing in the statement of the theorem is this common multiplicity.
Corollary of Clifford's theorem.
A corollary of Clifford's theorem, which is often exploited, is that the irreducible character χ appearing in the theorem is induced from an irreducible character of the inertial subgroup "IG"(μ). If, for example, the irreducible character χ is primitive (that is, χ is not induced from any proper subgroup of "G"), then "G" = "IG"(μ) and χN = "e"μ. A case where this property of primitive characters is used particularly frequently is when "N" is Abelian and χ is faithful (that is, its kernel contains just the identity element). In that case, μ is linear, "N" is represented by scalar matrices in any representation affording character χ and "N" is thus contained in the center of "G". For example, if "G" is the symmetric group "S"4, then "G" has a faithful complex irreducible character χ of degree "3." There is an Abelian normal subgroup "N" of order "4" (a Klein "4"-subgroup) which is not contained in the center of "G". Hence χ is induced from a character of a proper subgroup of "G" containing "N." The only possibility is that χ is induced from a linear character of a Sylow "2"-subgroup of "G".
Further developments.
Clifford's theorem has led to a branch of representation theory in its own right, now known as Clifford theory. This is particularly relevant to the representation theory of finite solvable groups, where normal subgroups usually abound. For more general finite groups, Clifford theory often allows representation-theoretic questions to be reduced to questions about groups that are close (in a sense which can be made precise) to being simple.
found a more precise version of this result for the restriction of irreducible unitary representations of locally compact groups to closed normal subgroups in what has become known as the "Mackey machine" or "Mackey normal subgroup analysis".
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abstract_algebra
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In mathematical finite group theory, the concept of regular "p"-group captures some of the more important properties of abelian "p"-groups, but is general enough to include most "small" "p"-groups. Regular "p"-groups were introduced by Phillip Hall (1934).
Definition.
A finite "p"-group "G" is said to be regular if any of the following equivalent , conditions are satisfied:
Examples.
Many familiar "p"-groups are regular:
However, many familiar "p"-groups are not regular:
Properties.
A "p"-group is regular if and only if every subgroup generated by two elements is regular.
Every subgroup and quotient group of a regular group is regular, but the direct product of regular groups need not be regular.
A 2-group is regular if and only if it is abelian. A 3-group with two generators is regular if and only if its derived subgroup is cyclic. Every "p"-group of odd order with cyclic derived subgroup is regular.
The subgroup of a "p"-group "G" generated by the elements of order dividing "p""k" is denoted Ω"k"("G") and regular groups are well-behaved in that Ω"k"("G") is precisely the set of elements of order dividing "p""k". The subgroup generated by all "p""k"-th powers of elements in "G" is denoted ℧"k"("G"). In a regular group, the index [G:℧"k"("G")] is equal to the order of Ω"k"("G"). In fact, commutators and powers interact in particularly simple ways . For example, given normal subgroups "M" and "N" of a regular "p"-group "G" and nonnegative integers "m" and "n", one has [℧"m"("M"),℧"n"("N")] = ℧"m"+"n"(["M","N"]).
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1984090
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abstract_algebra
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Algebraic structure
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod "p" when "p" is a prime number.
The "order" of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number "p" and every positive integer "k" there are fields of order "p""k", all of which are isomorphic.
Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.
Properties.
A finite field is a finite set that is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms.
The number of elements of a finite field is called its "order" or, sometimes, its "size". A finite field of order "q" exists if and only if "q" is a prime power "p""k" (where "p" is a prime number and "k" is a positive integer). In a field of order "p""k", adding "p" copies of any element always results in zero; that is, the characteristic of the field is "p".
If "q" = "p""k", all fields of order "q" are isomorphic (see ' below). Moreover, a field cannot contain two different finite subfields with the same order. One may therefore identify all finite fields with the same order, and they are unambiguously denoted $\mathbb{F}_{q}$, F"'"q" or GF("q"), where the letters GF stand for "Galois field".
In a finite field of order "q", the polynomial "Xq" − "X" has all "q" elements of the finite field as roots. The non-zero elements of a finite field form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. (In general there will be several primitive elements for a given field.)
The simplest examples of finite fields are the fields of prime order: for each prime number "p", the prime field of order "p" may be constructed as the integers modulo "p", Z / "p"Z.
The elements of the prime field of order "p" may be represented by integers in the range 0, ..., "p" − 1. The sum, the difference and the product are the remainder of the division by "p" of the result of the corresponding integer operation. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see "").
Let "F" be a finite field. For any element "x" in "F" and any integer "n", denote by "n" ⋅ "x" the sum of "n" copies of "x". The least positive "n" such that "n" ⋅ 1 = 0 is the characteristic "p" of the field. This allows defining a multiplication ("k", "x") ↦ "k" ⋅ "x" of an element "k" of GF("p") by an element "x" of "F" by choosing an integer representative for "k". This multiplication makes "F" into a GF("p")-vector space. It follows that the number of elements of "F" is "p""n" for some integer "n".
The identity
<math display="block" id="powersum">(x+y)^p=x^p+y^p$
(sometimes called the freshman's dream) is true in a field of characteristic "p". This follows from the binomial theorem, as each binomial coefficient of the expansion of ("x" + "y")"p", except the first and the last, is a multiple of "p".
By Fermat's little theorem, if "p" is a prime number and "x" is in the field GF("p") then "x""p" = "x". This implies the equality
<math display="block">X^p-X=\prod_{a\in \mathrm{GF}(p)} (X-a)$
for polynomials over GF("p"). More generally, every element in GF("p""n") satisfies the polynomial equation "x""p""n" − "x" = 0.
Any finite field extension of a finite field is separable and simple. That is, if "E" is a finite field and "F" is a subfield of "E", then "E" is obtained from "F" by adjoining a single element whose minimal polynomial is separable. To use a jargon, finite fields are perfect.
A more general algebraic structure that satisfies all the other axioms of a field, but whose multiplication is not required to be commutative, is called a division ring (or sometimes "skew field"). By Wedderburn's little theorem, any finite division ring is commutative, and hence is a finite field.
Existence and uniqueness.
Let "q" = "pn" be a prime power, and "F" be the splitting field of the polynomial
<math display="block">P = X^q-X$
over the prime field GF("p"). This means that "F" is a finite field of lowest order, in which "P" has "q" distinct roots (the formal derivative of "P" is "P"′ = −1, implying that gcd("P", "P"′) = 1, which in general implies that the splitting field is a separable extension of the original). The above identity shows that the sum and the product of two roots of "P" are roots of "P", as well as the multiplicative inverse of a root of "P". In other words, the roots of "P" form a field of order "q", which is equal to "F" by the minimality of the splitting field.
The uniqueness up to isomorphism of splitting fields implies thus that all fields of order "q" are isomorphic. Also, if a field "F" has a field of order "q" = "p""k" as a subfield, its elements are the "q" roots of "X""q" − "X", and "F" cannot contain another subfield of order "q".
In summary, we have the following classification theorem first proved in 1893 by E. H. Moore:
"The order of a finite field is a prime power. For every prime power" "q" "there are fields of order" "q", "and they are all isomorphic. In these fields, every element satisfies"
<math display="block">x^q=x,$
"and the polynomial" "Xq" − "X" "factors as"
<math display="block">X^q-X= \prod_{a\in F} (X-a).$
It follows that GF("pn") contains a subfield isomorphic to GF("p""m") if and only if "m" is a divisor of "n"; in that case, this subfield is unique. In fact, the polynomial "X""p""m" − "X" divides "X""p""n" − "X" if and only if "m" is a divisor of "n".
Explicit construction.
Non-prime fields.
Given a prime power "q" = "p""n" with "p" prime and "n" > 1, the field GF("q") may be explicitly constructed in the following way. One first chooses an irreducible polynomial "P" in GF("p")["X"] of degree "n" (such an irreducible polynomial always exists). Then the quotient ring
<math display="block">\mathrm{GF}(q) = \mathrm{GF}(p)[X]/(P)$
of the polynomial ring GF("p")["X"] by the ideal generated by "P" is a field of order "q".
More explicitly, the elements of GF("q") are the polynomials over GF("p") whose degree is strictly less than "n". The addition and the subtraction are those of polynomials over GF("p"). The product of two elements is the remainder of the Euclidean division by "P" of the product in GF("p")["X"].
The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see "".
However, with this representation, elements of GF("q") may be difficult to distinguish from the corresponding polynomials. Therefore, it is common to give a name, commonly "α" to the element of GF("q") that corresponds to the polynomial "X". So, the elements of GF("q") become polynomials in "α", where "P"("α") = 0, and, when one encounters a polynomial in "α" of degree greater of equal to "n" (for example after a multiplication), one knows that one has to use the relation "P"("α") = 0 to reduce its degree (it is what Euclidean division is doing).
Except in the construction of GF(4), there are several possible choices for "P", which produce isomorphic results. To simplify the Euclidean division, one commonly chooses for "P" a polynomial of the form
<math display="block">X^n + aX + b,$
which make the needed Euclidean divisions very efficient. However, for some fields, typically in characteristic 2, irreducible polynomials of the form "Xn" + "aX" + "b" may not exist. In characteristic 2, if the polynomial "X""n" + "X" + 1 is reducible, it is recommended to choose "X""n" + "X""k" + 1 with the lowest possible "k" that makes the polynomial irreducible. If all these trinomials are reducible, one chooses "pentanomials" "X""n" + "X""a" + "X""b" + "X""c" + 1, as polynomials of degree greater than 1, with an even number of terms, are never irreducible in characteristic 2, having 1 as a root.
A possible choice for such a polynomial is given by Conway polynomials. They ensure a certain compatibility between the representation of a field and the representations of its subfields.
In the next sections, we will show how the general construction method outlined above works for small finite fields.
Field with four elements.
The smallest non-prime field is the field with four elements, which is commonly denoted GF(4) or $\mathbb{F}_4.$ It consists of the four elements 0, 1, "α", 1 + "α" such that "α"2 = 1 + "α", 1 ⋅ "α" = "α" ⋅ 1 = "α", "x" + "x" = 0, and "x" ⋅ 0 = 0 ⋅ "x" = 0, for every "x" ∈ GF(4), the other operation results being easily deduced from the distributive law. See below for the complete operation tables.
This may be deduced as follows from the results of the preceding section.
Over GF(2), there is only one irreducible polynomial of degree 2:
<math display="block">X^2+X+1$
Therefore, for GF(4) the construction of the preceding section must involve this polynomial, and
<math display="block">\mathrm{GF}(4) = \mathrm{GF}(2)[X]/(X^2+X+1).$
Let "α" denote a root of this polynomial in GF(4). This implies that
"α"2 = 1 + "α",
and that "α" and 1 + "α" are the elements of GF(4) that are not in GF(2). The tables of the operations in GF(4) result from this, and are as follows:
A table for subtraction is not given, because subtraction is identical to addition, as is the case for every field of characteristic 2.
In the third table, for the division of "x" by "y", the values of "x" must be read in the left column, and the values of "y" in the top row. (Because 0 ⋅ "z" = 0 for every z in every ring the division by 0 has to remain undefined.) From the tables, it can be seen that the additive structure of GF(4) is isomorphic to the Klein four-group, while the non-zero multiplicative structure is isomorphic to the group Z3.
The map
<math display="block"> \varphi:x \mapsto x^2$
is the non-trivial field automorphism, called the Frobenius automorphism, which sends "α" into the second root 1 + "α" of the above mentioned irreducible polynomial "X"2 + "X" + 1.
GF("p"2) for an odd prime "p".
For applying the above general construction of finite fields in the case of GF("p"2), one has to find an irreducible polynomial of degree 2. For "p" = 2, this has been done in the preceding section. If "p" is an odd prime, there are always irreducible polynomials of the form "X"2 − "r", with "r" in GF("p").
More precisely, the polynomial "X"2 − "r" is irreducible over GF("p") if and only if "r" is a quadratic non-residue modulo "p" (this is almost the definition of a quadratic non-residue). There are quadratic non-residues modulo "p". For example, 2 is a quadratic non-residue for "p" = 3, 5, 11, 13, ..., and 3 is a quadratic non-residue for "p" = 5, 7, 17, ... If "p" ≡ 3 mod 4, that is "p" = 3, 7, 11, 19, ..., one may choose −1 ≡ "p" − 1 as a quadratic non-residue, which allows us to have a very simple irreducible polynomial "X"2 + 1.
Having chosen a quadratic non-residue "r", let "α" be a symbolic square root of "r", that is, a symbol that has the property "α"2 = "r", in the same way that the complex number "i" is a symbolic square root of −1. Then, the elements of GF("p"2) are all the linear expressions
<math display="block">a+b\alpha,$
with "a" and "b" in GF("p"). The operations on GF("p"2) are defined as follows (the operations between elements of GF("p") represented by Latin letters are the operations in GF("p")):
-(a+b\alpha)&=-a+(-b)\alpha\\
(a+b\alpha)+(c+d\alpha)&=(a+c)+(b+d)\alpha\\
(a+b\alpha)(c+d\alpha)&=(ac + rbd)+ (ad+bc)\alpha\\
(a+b\alpha)^{-1}&=a(a^2-rb^2)^{-1}+(-b)(a^2-rb^2)^{-1}\alpha
\end{align}$
GF(8) and GF(27).
The polynomial
<math display="block">X^3-X-1$
is irreducible over GF(2) and GF(3), that is, it is irreducible modulo 2 and 3 (to show this, it suffices to show that it has no root in GF(2) nor in GF(3)). It follows that the elements of GF(8) and GF(27) may be represented by expressions
<math display="block">a+b\alpha+c\alpha^2,$
where "a", "b", "c" are elements of GF(2) or GF(3) (respectively), and "α" is a symbol such that
<math display="block">\alpha^3=\alpha+1.$
The addition, additive inverse and multiplication on GF(8) and GF(27) may thus be defined as follows; in following formulas, the operations between elements of GF(2) or GF(3), represented by Latin letters, are the operations in GF(2) or GF(3), respectively:
<math display="block">
-(a+b\alpha+c\alpha^2)&=-a+(-b)\alpha+(-c)\alpha^2 \qquad\text{(for } \mathrm{GF}(8), \text{this operation is the identity)}\\
(a+b\alpha+c\alpha^2)+(d+e\alpha+f\alpha^2)&=(a+d)+(b+e)\alpha+(c+f)\alpha^2\\
(a+b\alpha+c\alpha^2)(d+e\alpha+f\alpha^2)&=(ad + bf+ce)+ (ae+bd+bf+ce+cf)\alpha+(af+be+cd+cf)\alpha^2
$
GF(16).
The polynomial
<math display="block">X^4+X+1$
is irreducible over GF(2), that is, it is irreducible modulo 2. It follows that the elements of GF(16) may be represented by expressions
<math display="block">a+b\alpha+c\alpha^2+d\alpha^3,$
where "a", "b", "c", "d" are either 0 or 1 (elements of GF(2)), and "α" is a symbol such that
<math display="block">\alpha^4=\alpha+1$
(that is, "α" is defined as a root of the given irreducible polynomial). As the characteristic of GF(2) is 2, each element is its additive inverse in GF(16). The addition and multiplication on GF(16) may be defined as follows; in following formulas, the operations between elements of GF(2), represented by Latin letters are the operations in GF(2).
<math display="block">
(a+b\alpha+c\alpha^2+d\alpha^3)+(e+f\alpha+g\alpha^2+h\alpha^3)&=(a+e)+(b+f)\alpha+(c+g)\alpha^2+(d+h)\alpha^3\\
+(af+be+bh+cg+df +ch+dg)\alpha\;+\\
&\quad\;(ag+bf+ce +ch+dg+dh)\alpha^2
+(ah+bg+cf+de +dh)\alpha^3
$
The field GF(16) has eight primitive elements (the elements that have all nonzero elements of GF(16) as integer powers). These elements are the four roots of "X"4 + "X" + 1 and their multiplicative inverses. In particular, "α" is a primitive element, and the primitive elements are "α""m" with "m" less than and coprime with 15 (that is, 1, 2, 4, 7, 8, 11, 13, 14).
Multiplicative structure.
The set of non-zero elements in GF("q") is an abelian group under the multiplication, of order "q" – 1. By Lagrange's theorem, there exists a divisor "k" of "q" – 1 such that "xk" = 1 for every non-zero "x" in GF("q"). As the equation "xk" = 1 has at most "k" solutions in any field, "q" – 1 is the lowest possible value for "k".
The structure theorem of finite abelian groups implies that this multiplicative group is cyclic, that is, all non-zero elements are powers of a single element. In summary:
"The multiplicative group of the non-zero elements in" GF("q") "is cyclic, and there exists an element" "a", "such that the" "q" – 1 "non-zero elements of" GF("q") "are" "a", "a"2, ..., "a""q"−2, "a""q"−1 = 1.
Such an element "a" is called a primitive element. Unless "q" = 2, 3, the primitive element is not unique. The number of primitive elements is "φ"("q" − 1) where "φ" is Euler's totient function.
The result above implies that "xq" = "x" for every "x" in GF("q"). The particular case where "q" is prime is Fermat's little theorem.
Discrete logarithm.
If "a" is a primitive element in GF("q"), then for any non-zero element "x" in "F", there is a unique integer "n" with 0 ≤ "n" ≤ "q" − 2 such that
"x" = "an".
This integer "n" is called the discrete logarithm of "x" to the base "a".
While "an" can be computed very quickly, for example using exponentiation by squaring, there is no known efficient algorithm for computing the inverse operation, the discrete logarithm. This has been used in various cryptographic protocols, see "Discrete logarithm" for details.
When the nonzero elements of GF("q") are represented by their discrete logarithms, multiplication and division are easy, as they reduce to addition and subtraction modulo "q" – 1. However, addition amounts to computing the discrete logarithm of "a""m" + "a""n". The identity
"a""m" + "a""n" = "a""n"("a""m"−"n" + 1)
allows one to solve this problem by constructing the table of the discrete logarithms of "a""n" + 1, called Zech's logarithms, for "n" = 0, ..., "q" − 2 (it is convenient to define the discrete logarithm of zero as being −∞).
Zech's logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table of the same size as the order of the field.
Roots of unity.
Every nonzero element of a finite field is a root of unity, as "x""q"−1 = 1 for every nonzero element of GF("q").
If "n" is a positive integer, an "n"th primitive root of unity is a solution of the equation "xn" = 1 that is not a solution of the equation "xm" = 1 for any positive integer "m" < "n". If "a" is a "n"th primitive root of unity in a field "F", then "F" contains all the "n" roots of unity, which are 1, "a", "a"2, ..., "a""n"−1.
The field GF("q") contains a "n"th primitive root of unity if and only if "n" is a divisor of "q" − 1; if "n" is a divisor of "q" − 1, then the number of primitive "n"th roots of unity in GF("q") is "φ"("n") (Euler's totient function). The number of "n"th roots of unity in GF("q") is gcd("n", "q" − 1).
In a field of characteristic "p", every ("np")th root of unity is also a "n"th root of unity. It follows that primitive ("np")th roots of unity never exist in a field of characteristic "p".
On the other hand, if "n" is coprime to "p", the roots of the "n"th cyclotomic polynomial are distinct in every field of characteristic "p", as this polynomial is a divisor of "X""n" − 1, whose discriminant "n""n" is nonzero modulo "p". It follows that the "n"th cyclotomic polynomial factors over GF("p") into distinct irreducible polynomials that have all the same degree, say "d", and that GF("p""d") is the smallest field of characteristic "p" that contains the "n"th primitive roots of unity.
Example: GF(64).
The field GF(64) has several interesting properties that smaller fields do not share: it has two subfields such that neither is contained in the other; not all generators (elements with minimal polynomial of degree 6 over GF(2)) are primitive elements; and the primitive elements are not all conjugate under the Galois group.
The order of this field being 26, and the divisors of 6 being 1, 2, 3, 6, the subfields of GF(64) are GF(2), GF(22) = GF(4), GF(23) = GF(8), and GF(64) itself. As 2 and 3 are coprime, the intersection of GF(4) and GF(8) in GF(64) is the prime field GF(2).
The union of GF(4) and GF(8) has thus 10 elements. The remaining 54 elements of GF(64) generate GF(64) in the sense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree 6 over GF(2). This implies that, over GF(2), there are exactly 9 = irreducible monic polynomials of degree 6. This may be verified by factoring "X"64 − "X" over GF(2).
The elements of GF(64) are primitive "n"th roots of unity for some "n" dividing 63. As the 3rd and the 7th roots of unity belong to GF(4) and GF(8), respectively, the 54 generators are primitive "n"th roots of unity for some "n" in {9, 21, 63}. Euler's totient function shows that there are 6 primitive 9th roots of unity, 12 primitive 21st roots of unity, and 36 primitive 63rd roots of unity. Summing these numbers, one finds again 54 elements.
By factoring the cyclotomic polynomials over GF(2), one finds that:
This shows that the best choice to construct GF(64) is to define it as GF(2)["X"] / ("X"6 + "X" + 1). In fact, this generator is a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division.
Frobenius automorphism and Galois theory.
In this section, "p" is a prime number, and "q" = "p""n" is a power of "p".
In GF("q"), the identity ("x" + "y")"p" = "xp" + "yp" implies that the map
<math display="block"> \varphi:x \mapsto x^p$
is a GF("p")-linear endomorphism and a field automorphism of GF("q"), which fixes every element of the subfield GF("p"). It is called the Frobenius automorphism, after Ferdinand Georg Frobenius.
Denoting by "φk" the composition of "φ" with itself "k" times, we have
<math display="block"> \varphi^k:x \mapsto x^{p^k}.$
It has been shown in the preceding section that "φ""n" is the identity. For 0 < "k" < "n", the automorphism "φ""k" is not the identity, as, otherwise, the polynomial
<math display="block">X^{p^k}-X$
would have more than "pk" roots.
There are no other GF("p")-automorphisms of GF("q"). In other words, GF("pn") has exactly "n" GF("p")-automorphisms, which are
<math display="block">\mathrm{Id}=\varphi^0, \varphi, \varphi^2, \ldots, \varphi^{n-1}.$
In terms of Galois theory, this means that GF("p""n") is a Galois extension of GF("p"), which has a cyclic Galois group.
The fact that the Frobenius map is surjective implies that every finite field is perfect.
Polynomial factorization.
If "F" is a finite field, a non-constant monic polynomial with coefficients in "F" is irreducible over "F", if it is not the product of two non-constant monic polynomials, with coefficients in "F".
As every polynomial ring over a field is a unique factorization domain, every monic polynomial over a finite field may be factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials.
There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite field. They are a key step for factoring polynomials over the integers or the rational numbers. At least for this reason, every computer algebra system has functions for factoring polynomials over finite fields, or, at least, over finite prime fields.
Irreducible polynomials of a given degree.
The polynomial
<math display="block">X^q-X$
factors into linear factors over a field of order "q". More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order "q".
This implies that, if "q" = "pn" then "Xq" − "X" is the product of all monic irreducible polynomials over GF("p"), whose degree divides "n". In fact, if "P" is an irreducible factor over GF("p") of "Xq" − "X", its degree divides "n", as its splitting field is contained in GF("p""n"). Conversely, if "P" is an irreducible monic polynomial over GF("p") of degree "d" dividing "n", it defines a field extension of degree "d", which is contained in GF("p""n"), and all roots of "P" belong to GF("p""n"), and are roots of "Xq" − "X"; thus "P" divides "Xq" − "X". As "Xq" − "X" does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it.
This property is used to compute the product of the irreducible factors of each degree of polynomials over GF("p"); see "Distinct degree factorization".
Number of monic irreducible polynomials of a given degree over a finite field.
The number "N"("q", "n") of monic irreducible polynomials of degree "n" over
GF("q") is given by
<math display="block">N(q,n)=\frac{1}{n}\sum_{d\mid n} \mu(d)q^{n/d},$
where "μ" is the Möbius function. This formula is an immediate consequence of the property of "X""q" − "X" above and the Möbius inversion formula.
By the above formula, the number of irreducible (not necessarily monic) polynomials of degree "n" over GF("q") is ("q" − 1)"N"("q", "n").
The exact formula implies the inequality
<math display="block">N(q,n)\geq\frac{1}{n} \left(q^n-\sum_{\ell\mid n, \ \ell \text{ prime}} q^{n/\ell}\right);$
this is sharp if and only if "n" is a power of some prime.
For every "q" and every "n", the right hand side is positive, so there is at least one irreducible polynomial of degree "n" over GF("q").
Applications.
In cryptography, the difficulty of the discrete logarithm problem in finite fields or in elliptic curves is the basis of several widely used protocols, such as the Diffie–Hellman protocol. For example, in 2014, a secure internet connection to Wikipedia involved the elliptic curve Diffie–Hellman protocol (ECDHE) over a large finite field. In coding theory, many codes are constructed as subspaces of vector spaces over finite fields.
Finite fields are used by many error correction codes, such as Reed–Solomon error correction code or BCH code. The finite field almost always has characteristic of 2, since computer data is stored in binary. For example, a byte of data can be interpreted as an element of GF(28). One exception is PDF417 bar code, which is GF(929). Some CPUs have special instructions that can be useful for finite fields of characteristic 2, generally variations of carry-less product.
Finite fields are widely used in number theory, as many problems over the integers may be solved by reducing them modulo one or several prime numbers. For example, the fastest known algorithms for polynomial factorization and linear algebra over the field of rational numbers proceed by reduction modulo one or several primes, and then reconstruction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.
Similarly many theoretical problems in number theory can be solved by considering their reductions modulo some or all prime numbers. See, for example, "Hasse principle". Many recent developments of algebraic geometry were motivated by the need to enlarge the power of these modular methods. Wiles' proof of Fermat's Last Theorem is an example of a deep result involving many mathematical tools, including finite fields.
The Weil conjectures concern the number of points on algebraic varieties over finite fields and the theory has many applications including exponential and character sum estimates.
Finite fields have widespread application in combinatorics, two well known examples being the definition of Paley Graphs and the related construction for Hadamard Matrices. In arithmetic combinatorics finite fields and finite field models are used extensively, such as in Szemerédi's theorem on arithmetic progressions.
Extensions.
Algebraic closure.
A finite field "F" is not algebraically closed: the polynomial
<math display="block">f(T) = 1+\prod_{\alpha \in F} (T-\alpha),$
has no roots in "F", since "f" ("α") = 1 for all "α" in "F".
Fix an algebraic closure $\overline{\mathbb{F}}_q$ of $\mathbb{F}_q$. The map $\varphi_q \colon \overline{\mathbb{F}}_q \to \overline{\mathbb{F}}_q$ sending each "x" to "x""q" is called the "q"th power Frobenius automorphism. The subfield of $\overline{\mathbb{F}}_q$ fixed by the "n"th iterate of "φ""q" is the set of zeros of the polynomial "x""q""n" − "x", which has distinct roots since its derivative in $\mathbb{F}_q[x]$ is −1, which is never zero. Therefore, that subfield has "q""n" elements, so it is the unique copy of $\mathbb{F}_{q^n}$ in $\overline{\mathbb{F}}_q$. Every finite extension of $\mathbb{F}_q$ in $\overline{\mathbb{F}}_q$ is this $\mathbb{F}_{q^n}$ for some "n", so
<math display="block">\overline{\mathbb{F}}_q = \bigcup_{n \ge 1} \mathbb{F}_{q^n}.$
The absolute Galois group of $\mathbb{F}_q$ is the profinite group
<math display="block">\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q) \simeq \varprojlim_n \operatorname{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q) \simeq \varprojlim_n (\mathbf{Z}/n\mathbf{Z}) = \widehat{\mathbf{Z}}.$
Like any infinite Galois group, $\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)$ may be equipped with the Krull topology, and then the isomorphisms just given are isomorphisms of topological groups.
The image of "φ""q" in the group $\operatorname{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q) \simeq \mathbf{Z}/n\mathbf{Z}$ is the generator 1, so "φ""q" corresponds to $1 \in \widehat{\mathbf{Z}}$. It follows that "φ""q" has infinite order and generates a dense subgroup of $\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)$, not the whole group, because the element $1 \in \widehat{\mathbf{Z}}$ has infinite order and generates the dense subgroup $\mathbf{Z} \subsetneqq \widehat{\mathbf{Z}}.$ One says that "φ""q" is a topological generator of $\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)$.
Quasi-algebraic closure.
Although finite fields are not algebraically closed, they are quasi-algebraically closed, which means that every homogeneous polynomial over a finite field has a non-trivial zero whose components are in the field if the number of its variables is more than its degree. This was a conjecture of Artin and Dickson proved by Chevalley (see "Chevalley–Warning theorem").
Wedderburn's little theorem.
A division ring is a generalization of field. Division rings are not assumed to be commutative. There are no non-commutative finite division rings: Wedderburn's little theorem states that all finite division rings are commutative, and hence are finite fields. This result holds even if we relax the associativity axiom to alternativity, that is, all finite alternative division rings are finite fields, by the Artin–Zorn theorem.
Notes.
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Mathematical statement of uniqueness, except for an equivalent structure (equivalence relation)Two mathematical objects a and b are called "equal up to an equivalence relation R"
This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count.
For example, "x is unique up to R" means that all objects x under consideration are in the same equivalence class with respect to the relation R.
Moreover, the equivalence relation R is often designated rather implicitly by a generating condition or transformation.
For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation R that relates two lists if one can be obtained by reordering (permuting) the other. As another example, the statement "the solution to an indefinite integral is sin("x"), up to addition of a constant" tacitly employs the equivalence relation R between functions, defined by "fRg" if the difference "f"−"g" is a constant function, and means that the solution and the function sin("x") are equal up to this R.
In the picture, "there are 4 partitions up to rotation" means that the set P has 4 equivalence classes with respect to R defined by "aRb" if b can be obtained from a by rotation; one representative from each class is shown in the bottom left picture part.
Equivalence relations are often used to disregard possible differences of objects, so "up to R" can be understood informally as "ignoring the same subtleties as R ignores".
In the factorization example, "up to ordering" means "ignoring the particular ordering".
Further examples include "up to isomorphism", "up to permutations", and "up to rotations", which are described in the Examples section.
In informal contexts, mathematicians often use the word modulo (or simply mod) for similar purposes, as in "modulo isomorphism".
Examples.
Tetris.
Consider the seven Tetris pieces (I, J, L, O, S, T, Z), known mathematically as the tetrominoes. If you consider all the possible rotations of these pieces — for example, if you consider the "I" oriented vertically to be distinct from the "I" oriented horizontally — then you find there are 19 distinct possible shapes to be displayed on the screen. (These 19 are the so-called "fixed" tetrominoes.) But if rotations are not considered distinct — so that we treat both "I vertically" and "I horizontally" indifferently as "I" — then there are only seven. We say that "there are seven tetrominoes, up to rotation". One could also say that "there are five tetrominoes, up to rotation and reflection", which accounts for the fact that L reflected gives J, and S reflected gives Z.
Eight queens.
In the eight queens puzzle, if the queens are considered to be distinct (e.g. if they are colored with eight different colors), then there are 3709440 distinct solutions. Normally, however, the queens are considered to be interchangeable, and one usually says "there are 3,709,440 / 8! = 92 unique solutions up to permutation of the queens", or that "there are 92 solutions modulo the names of the queens", signifying that two different arrangements of the queens are considered equivalent if the queens have been permuted, as long as the set of occupied squares remains the same.
If, in addition to treating the queens as identical, rotations and reflections of the board were allowed, we would have only 12 distinct solutions "up to symmetry and the naming of the queens". For more, see .
Polygons.
The regular n-gon, for a fixed n, is unique up to similarity. In other words, the "similarity" equivalence relation over the regular n-gons (for a fixed n) has only one equivalence class; it is impossible to produce two regular n-gons which are not similar to each other.
Group theory.
In group theory, one may have a group G acting on a set X, in which case, one might say that two elements of X are equivalent "up to the group action"—if they lie in the same orbit.
Another typical example is the statement that "there are two different groups of order 4 up to isomorphism", or "modulo isomorphism, there are two groups of order 4". This means that, if one considers isomorphic groups "equivalent", there are only two equivalence classes of groups of order 4.
Nonstandard analysis.
A hyperreal x and its standard part st("x") are equal up to an infinitesimal difference.
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Five sporadic simple groups
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups "M"11, "M"12, "M"22, "M"23 and "M"24 introduced by Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered.
Sometimes the notation "M"8 "M"9, "M"10, "M"20 and "M"21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid "M"13 acting on 13 points. "M"21 is simple, but is not a sporadic group, being isomorphic to PSL(3,4).
History.
introduced the group "M"12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group "M"24, giving its order. In he gave further details, including explicit generating sets for his groups, but it was not easy to see from his arguments that the groups generated are not just alternating groups, and for several years the existence of his groups was controversial. even published a paper mistakenly claiming to prove that "M"24 does not exist, though shortly afterwards in he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. Witt (1938a, 1938b) finally removed the doubts about the existence of these groups, by constructing them as successive transitive extensions of permutation groups, as well as automorphism groups of Steiner systems.
After the Mathieu groups no new sporadic groups were found until 1965, when the group J1 was discovered.
Multiply transitive groups.
Mathieu was interested in finding multiply transitive permutation groups, which will now be defined. For a natural number "k", a permutation group "G" acting on "n" points is "k"-transitive if, given two sets of points "a"1, ... "a""k" and "b"1, ... "b""k" with the property that all the "a""i" are distinct and all the "b""i" are distinct, there is a group element "g" in "G" which maps "a""i" to "b""i" for each "i" between 1 and "k". Such a group is called sharply "k"-transitive if the element "g" is unique (i.e. the action on "k"-tuples is regular, rather than just transitive).
"M"24 is 5-transitive, and "M"12 is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of "m" points, and accordingly of lower transitivity ("M"23 is 4-transitive, etc.). These are the only two 5-transitive groups that are neither symmetric groups nor alternating groups .
The only 4-transitive groups are the symmetric groups "S""k" for "k" at least 4, the alternating groups "A""k" for "k" at least 6, and the Mathieu groups "M"24, "M"23, "M"12, and "M"11. The full proof requires the classification of finite simple groups, but some special cases have been known for much longer.
It is a classical result of Jordan that the symmetric and alternating groups (of degree "k" and "k" + 2 respectively), and "M"12 and "M"11 are the only "sharply" "k"-transitive permutation groups for "k" at least 4.
Important examples of multiply transitive groups are the 2-transitive groups and the Zassenhaus groups. The Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL(2,F"q"), which is sharply 3-transitive (see cross ratio) on $q+1$ elements.
Constructions of the Mathieu groups.
The Mathieu groups can be constructed in various ways.
Permutation groups.
"M"12 has a simple subgroup of order 660, a maximal subgroup. That subgroup is isomorphic to the projective special linear group PSL2(F11) over the field of 11 elements. With −1 written as a and infinity as b, two standard generators are (0123456789a) and (0b)(1a)(25)(37)(48)(69). A third generator giving "M"12 sends an element "x" of F11 to 4"x"2 − 3"x"7; as a permutation that is (26a7)(3945).
This group turns out not to be isomorphic to any member of the infinite families of finite simple groups and is called sporadic. "M"11 is the stabilizer of a point in "M"12, and turns out also to be a sporadic simple group. "M"10, the stabilizer of two points, is not sporadic, but is an almost simple group whose commutator subgroup is the alternating group A6. It is thus related to the exceptional outer automorphism of A6. The stabilizer of 3 points is the projective special unitary group PSU(3,22), which is solvable. The stabilizer of 4 points is the quaternion group.
Likewise, "M"24 has a maximal simple subgroup of order 6072 isomorphic to PSL2(F23). One generator adds 1 to each element of the field (leaving the point "N" at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)("N"), and the other is the order reversing permutation, (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving "M"24 sends an element "x" of F23 to 4"x"4 − 3"x"15 (which sends perfect squares via $ x^4 $ and non-perfect squares via $ 7 x^4$); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).
The stabilizers of 1 and 2 points, "M"23 and "M"22 also turn out to be sporadic simple groups. The stabilizer of 3 points is simple and isomorphic to the projective special linear group PSL3(4).
These constructions were cited by . ascribe the permutations to Mathieu.
Automorphism groups of Steiner systems.
There exists up to equivalence a unique "S"(5,8,24) Steiner system W24 (the Witt design). The group "M"24 is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups "M"23 and "M"22 are defined to be the stabilizers of a single point and two points respectively.
Similarly, there exists up to equivalence a unique "S"(5,6,12) Steiner system W12, and the group "M"12 is its automorphism group. The subgroup "M"11 is the stabilizer of a point.
"W"12 can be constructed from the affine geometry on the vector space F3 × F3, an "S"(2,3,9) system.
An alternative construction of "W"12 is the 'Kitten' of .
An introduction to a construction of "W"24 via the Miracle Octad Generator of R. T. Curtis and Conway's analog for "W"12, the miniMOG, can be found in the book by Conway and Sloane.
Automorphism groups on the Golay code.
The group "M"24 is the permutation automorphism group of the extended binary Golay code "W", i.e., the group of permutations on the 24 coordinates that map "W" to itself. All the Mathieu groups can be constructed as groups of permutations on the binary Golay code.
"M"12 has index 2 in its automorphism group, and "M"12:2 happens to be isomorphic to a subgroup of "M"24. "M"12 is the stabilizer of a dodecad, a codeword of 12 1's; "M"12:2 stabilizes a partition into 2 complementary dodecads.
There is a natural connection between the Mathieu groups and the larger Conway groups, because the Leech lattice was constructed on the binary Golay code and in fact both lie in spaces of dimension 24. The Conway groups in turn are found in the Monster group. Robert Griess refers to the 20 sporadic groups found in the Monster as the Happy Family, and to the Mathieu groups as the first generation.
Dessins d'enfants.
The Mathieu groups can be constructed via dessins d'enfants, with the dessin associated to "M"12 suggestively called "Monsieur Mathieu" by .
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Theorem in group theory
The Schur–Zassenhaus theorem is a theorem in group theory which states that if $G$ is a finite group, and $N$ is a normal subgroup whose order is coprime to the order of the quotient group $G/N$, then $G$ is a semidirect product (or split extension) of $N$ and $G/N$. An alternative statement of the theorem is that any normal Hall subgroup $N$ of a finite group $G$ has a complement in $G$. Moreover if either $N$ or $G/N$ is solvable then the Schur–Zassenhaus theorem also states that all complements of $N$ in $G$ are conjugate. The assumption that either $N$ or $G/N$ is solvable can be dropped as it is always satisfied, but all known proofs of this require the use of the much harder Feit–Thompson theorem.
The Schur–Zassenhaus theorem at least partially answers the question: "In a composition series, how can we classify groups with a certain set of composition factors?" The other part, which is where the composition factors do not have coprime orders, is tackled in extension theory.
History.
The Schur–Zassenhaus theorem was introduced by Zassenhaus (1937, 1958, Chapter IV, section 7). Theorem 25, which he credits to Issai Schur, proves the existence of a complement, and theorem 27 proves that all complements are conjugate under the assumption that $N$ or $G/N$ is solvable. It is not easy to find an explicit statement of the existence of a complement in Schur's published works, though the results of Schur (1904, 1907) on the Schur multiplier imply the existence of a complement in the special case when the normal subgroup is in the center. Zassenhaus pointed out that the Schur–Zassenhaus theorem for non-solvable groups would follow if all groups of odd order are solvable, which was later proved by Feit and Thompson. Ernst Witt showed that it would also follow from the Schreier conjecture (see Witt (1998, p.277) for Witt's unpublished 1937 note about this), but the Schreier conjecture has only been proved using the classification of finite simple groups, which is far harder than the Feit–Thompson theorem.
Examples.
If we do not impose the coprime condition, the theorem is not true: consider for example the cyclic group $C_4$ and its normal subgroup $C_2$. Then if $C_4$ were a semidirect product of $C_2$ and $C_4 / C_2 \cong C_2$ then $C_4$ would have to contain two elements of order 2, but it only contains one. Another way to explain this impossibility of splitting $C_4$ (i.e. expressing it as a semidirect product) is to observe that the automorphisms of $C_2$ are the trivial group, so the only possible [semi]direct product of $C_2$ with itself is a direct product (which gives rise to the Klein four-group, a group that is non-isomorphic with $C_4$).
An example where the Schur–Zassenhaus theorem does apply is the symmetric group on 3 symbols, $S_3$, which has a normal subgroup of order 3 (isomorphic with $C_3$) which in turn has index 2 in $S_3$ (in agreement with the theorem of Lagrange), so $S_3 / C_3 \cong C_2$. Since 2 and 3 are relatively prime, the Schur–Zassenhaus theorem applies and $S_3 \cong C_3 \rtimes C_2$. Note that the automorphism group of $C_3$ is $C_2$ and the automorphism of $C_3$ used in the semidirect product that gives rise to $S_3$ is the non-trivial automorphism that permutes the two non-identity elements of $C_3$. Furthermore, the three subgroups of order 2 in $S_3$ (any of which can serve as a complement to $C_3$ in $S_3$) are conjugate to each other.
The non-triviality of the (additional) conjugacy conclusion can be illustrated with the Klein four-group $V$ as the non-example. Any of the three proper subgroups of $V$ (all of which have order 2) is normal in $V$; fixing one of these subgroups, any of the other two remaining (proper) subgroups complements it in $V$, but none of these three subgroups of $V$ is a conjugate of any other one, because $V$ is abelian.
The quaternion group has normal subgroups of order 4 and 2 but is not a [semi]direct product. Schur's papers at the beginning of the 20th century introduced the notion of central extension to address examples such as $C_4$ and the quaternions.
Proof.
The existence of a complement to a normal Hall subgroup "H" of a finite group "G" can be proved in the following steps:
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In group theory, a subfield of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.
A cycle is the set of powers of a given group element "a", where "an", the "n"-th power of an element "a" is defined as the product of "a" multiplied by itself "n" times. The element "a" is said to "generate" the cycle. In a finite group, some non-zero power of "a" must be the group identity, "e"; the lowest such power is the order of the cycle, the number of distinct elements in it. In a cycle graph, the cycle is represented as a polygon, with the vertices representing the group elements, and the connecting lines indicating that all elements in that polygon are members of the same cycle.
Cycles.
Cycles can overlap, or they can have no element in common but the identity. The cycle graph displays each interesting cycle as a polygon.
If "a" generates a cycle of order 6 (or, more shortly, "has" order 6), then "a"6 = "e". Then the set of powers of "a"2, {"a"2, "a"4, "e"} is a cycle, but this is really no new information. Similarly, "a"5 generates the same cycle as "a" itself.
So, only the "primitive" cycles need be considered, namely those that are not subsets of another cycle. Each of these is generated by some "primitive element", "a". Take one point for each element of the original group. For each primitive element, connect "e" to "a", "a" to "a"2, ..., "a""n"−1 to "a""n", etc., until "e" is reached. The result is the cycle graph.
When "a"2 = "e", "a" has order 2 (is an involution), and is connected to "e" by two edges. Except when the intent is to emphasize the two edges of the cycle, it is typically drawn as a single line between the two elements.
Properties.
As an example of a group cycle graph, consider the dihedral group Dih4. The multiplication table for this group is shown on the left, and the cycle graph is shown on the right with "e" specifying the identity element.
Notice the cycle {"e", "a", "a"2, "a"3} in the multiplication table, with "a"4 = "e". The inverse "a"−1 = "a"3 is also a generator of this cycle: ("a"3)2 = "a"2, ("a"3)3 = "a", and ("a"3)4 = "e". Similarly, any cycle in any group has at least two generators, and may be traversed in either direction. More generally, the number of generators of a cycle with "n" elements is given by the Euler φ function of "n", and any of these generators may be written as the first node in the cycle (next to the identity "e"); or more commonly the nodes are left unmarked. Two distinct cycles cannot intersect in a generator.
Cycles that contain a non-prime number of elements have cyclic subgroups that are not shown in the graph. For the group Dih4 above, we could draw a line between "a"2 and "e" since ("a"2)2 = "e", but since "a"2 is part of a larger cycle, this is not an edge of the cycle graph.
There can be ambiguity when two cycles share a non-identity element. For example, the 8-element quaternion group has cycle graph shown at right. Each of the elements in the middle row when multiplied by itself gives −1 (where 1 is the identity element). In this case we may use different colors to keep track of the cycles, although symmetry considerations will work as well.
As noted earlier, the two edges of a 2-element cycle are typically represented as a single line.
The inverse of an element is the node symmetric to it in its cycle, with respect to the reflection which fixes the identity.
History.
Cycle graphs were investigated by the number theorist Daniel Shanks in the early 1950s as a tool to study multiplicative groups of residue classes. Shanks first published the idea in the 1962 first edition of his book "Solved and Unsolved Problems in Number Theory". In the book, Shanks investigates which groups have isomorphic cycle graphs and when a cycle graph is planar. In the 1978 second edition, Shanks reflects on his research on class groups and the development of the baby-step giant-step method: The cycle graphs have proved to be useful when working with finite Abelian groups; and I have used them frequently in finding my way around an intricate structure [77, p. 852], in obtaining a wanted multiplicative relation [78, p. 426], or in isolating some wanted subgroup [79].
Cycle graphs are used as a pedagogical tool in Nathan Carter's 2009 introductory textbook "Visual Group Theory".
Graph characteristics of particular group families.
Certain group types give typical graphs:
Cyclic groups Z"n", order "n", is a single cycle graphed simply as an "n"-sided polygon with the elements at the vertices:
When "n" is a prime number, groups of the form (Z"n")"m" will have ("n""m" − 1)/("n" − 1) "n"-element cycles sharing the identity element:
Dihedral groups Dih"n", order 2"n" consists of an "n"-element cycle and "n" 2-element cycles:
Dicyclic groups, Dicn = Q4"n", order 4"n":
Other direct products:
Symmetric groups – The symmetric group S"n" contains, for any group of order "n", a subgroup isomorphic to that group. Thus the cycle graph of every group of order "n" will be found in the cycle graph of S"n".<br>
See example:
Example: Subgroups of the full octahedral group.
The full octahedral group is the direct product $S_4 \times Z_2$ of the symmetric group S4 and the cyclic group Z2.<br>
Its order is 48, and it has subgroups of every order that divides 48.
In the examples below nodes that are related to each other are placed next to each other,<br>
so these are not the simplest possible cycle graphs for these groups (like those on the right).
Like all graphs a cycle graph can be represented in different ways to emphasize different properties. The two representations of the cycle graph of S4 are an example of that.
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3D symmetry group
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4.
Details.
Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system.
Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these circles meet at order 2 and 3 gyration points.
Chiral tetrahedral symmetry.
T, 332, [3,3]+, or 23, of order 12 – chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry "D"2 or 222, with in addition four 3-fold axes, centered "between" the three orthogonal directions. This group is isomorphic to "A"4, the alternating group on 4 elements; in fact it is the group of even permutations of the four 3-fold axes: e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23).
The conjugacy classes of T are:
The rotations by 180°, together with the identity, form a normal subgroup of type Dih2, with quotient group of type Z3. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.
A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group "G" and a divisor "d" of |"G"|, there does not necessarily exist a subgroup of "G" with order "d": the group "G" = A4 has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be C6 or D3, but neither applies.
Achiral tetrahedral symmetry.
Td, *332, [3,3] or 43m, of order 24 – achiral or full tetrahedral symmetry, also known as the (2,3,3) triangle group. This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 (4) axes. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion. See also the isometries of the regular tetrahedron.
The conjugacy classes of Td are:
Pyritohedral symmetry.
Th, 3*2, [4,3+] or m3, of order 24 – pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S6 (3) axes, and there is a central inversion symmetry. Th is isomorphic to T × Z2: every element of Th is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D2h (that of a cuboid), of type Dih2 × Z2 = Z2 × Z2 × Z2. It is the direct product of the normal subgroup of T (see above) with C"i". The quotient group is the same as above: of type Z3. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.
It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a pyritohedron, which is extremely similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes.
The conjugacy classes of Th include those of T, with the two classes of 4 combined, and each with inversion:
Solids with chiral tetrahedral symmetry.
The Icosahedron colored as a snub tetrahedron has chiral symmetry.
Citations.
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689115
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abstract_algebra
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Concept in abstract algebra
In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime "p" and positive integer "n" there are exactly two (up to isomorphism) extraspecial groups of order "p"1+2"n". Extraspecial groups often occur in centralizers of involutions. The ordinary character theory of extraspecial groups is well understood.
Definition.
Recall that a finite group is called a "p"-group if its order is a power of a prime "p".
A "p"-group "G" is called extraspecial if its center "Z" is cyclic of order "p", and the quotient "G"/"Z" is a non-trivial elementary abelian "p"-group.
Extraspecial groups of order "p"1+2"n" are often denoted by the symbol "p"1+2"n". For example, 21+24 stands for an extraspecial group of order 225.
Classification.
Every extraspecial "p"-group has order "p"1+2"n" for some positive integer "n", and conversely for each such number there are exactly two extraspecial groups up to isomorphism. A central product of two extraspecial "p"-groups is extraspecial, and every extraspecial group can be written as a central product of extraspecial groups of order "p"3. This reduces the classification of extraspecial groups to that of extraspecial groups of order "p"3. The classification is often presented differently in the two cases "p" odd and "p" = 2, but a uniform presentation is also possible.
"p" odd.
There are two extraspecial groups of order "p"3, which for "p" odd are given by
If "n" is a positive integer there are two extraspecial groups of order "p"1+2"n", which for "p" odd are given by
The two extraspecial groups of order "p"1+2"n" are most easily distinguished by the fact that one has all elements of order at most "p" and the other has elements of order "p"2.
"p" = 2.
There are two extraspecial groups of order 8 = "2"3, which are given by
If "n" is a positive integer there are two extraspecial groups of order "2"1+2"n", which are given by
The two extraspecial groups "G" of order "2"1+2"n" are most easily distinguished as follows. If "Z" is the center, then "G"/"Z" is a vector space over the field with 2 elements. It has a quadratic form "q", where "q" is 1 if the lift of an element has order 4 in "G", and 0 otherwise. Then the Arf invariant of this quadratic form can be used to distinguish the two extraspecial groups. Equivalently, one can distinguish the groups by counting the number of elements of order 4.
All "p".
A uniform presentation of the extraspecial groups of order "p"1+2"n" can be given as follows. Define the two groups:
"M"("p") and "N"("p") are non-isomorphic extraspecial groups of order "p"3 with center of order "p" generated by "c". The two non-isomorphic extraspecial groups of order "p"1+2"n" are the central products of either "n" copies of "M"("p") or "n"−1 copies of "M"("p") and 1 copy of "N"("p"). This is a special case of a classification of "p"-groups with cyclic centers and simple derived subgroups given in .
Character theory.
If "G" is an extraspecial group of order "p"1+2"n", then its irreducible complex representations are given as follows:
Examples.
It is quite common for the centralizer of an involution in a finite simple group to contain a normal extraspecial subgroup. For example, the centralizer of an involution of type 2B in the monster group has structure 21+24.Co1, which means that it has a normal extraspecial subgroup of order 21+24, and the quotient is one of the Conway groups.
Generalizations.
Groups whose center, derived subgroup, and Frattini subgroup are all equal are called special groups. Infinite special groups whose derived subgroup has order "p" are also called extraspecial groups. The classification of countably infinite extraspecial groups is very similar to the finite case, , but for larger cardinalities even basic properties of the groups depend on delicate issues of set theory, some of which are exposed in . The nilpotent groups whose center is cyclic and derived subgroup has order "p" and whose conjugacy classes are at most countably infinite are classified in . Finite groups whose derived subgroup has order "p" are classified in .
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abstract_algebra
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Group of symmetries of an n-dimensional hypercube
In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter n, the dimension of the hypercube.
As a Coxeter group it is of type B"n" = C"n", and as a Weyl group it is associated to the symplectic groups and with the orthogonal groups in odd dimensions. As a wreath product it is $S_2 \wr S_n$ where Sn is the symmetric group of degree n. As a permutation group, the group is the signed symmetric group of permutations "π" either of the set } or of the set } such that for all i. As a matrix group, it can be described as the group of "n" × "n" orthogonal matrices whose entries are all integers. Equivalently, this is the set of "n" × "n" matrices with entries only 0, 1, or –1, which are invertible, and which have exactly one non-zero entry in each row or column. The representation theory of the hyperoctahedral group was described by according to .
In three dimensions, the hyperoctahedral group is known as "O" × "S"2 where "O" ≅ "S"4 is the octahedral group, and "S"2 is a symmetric group (here a cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry, named after the regular octahedron, or 3-orthoplex. In 4-dimensions it is called a hexadecachoric symmetry, after the regular 16-cell, or 4-orthoplex. In two dimensions, the hyperoctahedral group structure is the abstract dihedral group of order eight, describing the symmetry of a square, or 2-orthoplex.
By dimension.
Hyperoctahedral groups can be named as Bn, a bracket notation, or as a Coxeter group graph:
Subgroups.
There is a notable index two subgroup, corresponding to the Coxeter group "D""n" and the symmetries of the demihypercube. Viewed as a wreath product, there are two natural maps from the hyperoctahedral group to the cyclic group of order 2: one map coming from "multiply the signs of all the elements" (in the "n" copies of $\{\pm 1\}$), and one map coming from the parity of the permutation. Multiplying these together yields a third map $C_n \to \{\pm 1\}$. The kernel of the first map is the Coxeter group $D_n.$ In terms of signed permutations, thought of as matrices, this third map is simply the determinant, while the first two correspond to "multiplying the non-zero entries" and "parity of the underlying (unsigned) permutation", which are not generally meaningful for matrices, but are in the case due to the coincidence with a wreath product.
The kernels of these three maps are all three index two subgroups of the hyperoctahedral group, as discussed in below, and their intersection is the derived subgroup, of index 4 (quotient the Klein 4-group), which corresponds to the rotational symmetries of the demihypercube.
In the other direction, the center is the subgroup of scalar matrices, {±1}; geometrically, quotienting out by this corresponds to passing to the projective orthogonal group.
In dimension 2 these groups completely describe the hyperoctahedral group, which is the dihedral group Dih4 of order 8, and is an extension 2.V (of the 4-group by a cyclic group of order 2). In general, passing to the subquotient (derived subgroup, mod center) is the symmetry group of the projective demihypercube.
The hyperoctahedral subgroup, Dn by dimension:
The chiral hyper-octahedral symmetry, is the direct subgroup, index 2 of hyper-octahedral symmetry.
Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension: These groups have "n" orthogonal mirrors in "n"-dimensions.
Homology.
The group homology of the hyperoctahedral group is similar to that of the symmetric group, and exhibits stabilization, in the sense of stable homotopy theory.
H1: abelianization.
The first homology group, which agrees with the abelianization, stabilizes at the Klein four-group, and is given by:
$H_1(C_n, \mathbf{Z}) = \begin{cases} 0 & n = 0\\
\mathbf{Z}/2 & n = 1\\
\mathbf{Z}/2 \times \mathbf{Z}/2 & n \geq 2 \end{cases}.$
This is easily seen directly: the $-1$ elements are order 2 (which is non-empty for $n\geq 1$), and all conjugate, as are the transpositions in $S_n$ (which is non-empty for $n\geq 2$), and these are two separate classes. These elements generate the group, so the only non-trivial abelianizations are to 2-groups, and either of these classes can be sent independently to $-1 \in \{\pm 1\},$ as they are two separate classes. The maps are explicitly given as "the product of the signs of all the elements" (in the "n" copies of $\{\pm 1\}$), and the sign of the permutation. Multiplying these together yields a third non-trivial map (the determinant of the matrix, which sends both these classes to $-1$), and together with the trivial map these form the 4-group.
H2: Schur multipliers.
The second homology groups, known classically as the Schur multipliers, were computed in .
They are:
$H_2(C_n,\mathbf{Z}) = \begin{cases}
0 & n = 0, 1\\
\mathbf{Z}/2 & n = 2\\
(\mathbf{Z}/2)^2 & n = 3\\
(\mathbf{Z}/2)^3 & n \geq 4 \end{cases}.$
Notes.
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abstract_algebra
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Mathematical Tool in Group Theory
Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group – such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center – can be discovered from its Cayley table.
A simple example of a Cayley table is the one for the group {1, −1} under ordinary multiplication:
History.
Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation "θ" "n" = 1". In that paper they were referred to simply as tables, and were merely illustrative – they came to be known as Cayley tables later on, in honour of their creator.
Structure and layout.
Because many Cayley tables describe groups that are not abelian, the product "ab" with respect to the group's binary operation is not guaranteed to be equal to the product "ba" for all "a" and "b" in the group. In order to avoid confusion, the convention is that the factor that labels the row (termed "nearer factor" by Cayley) comes first, and that the factor that labels the column (or "further factor") is second. For example, the intersection of row "a" and column "b" is "ab" and not "ba", as in the following example:
Properties and uses.
Commutativity.
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and inspection of the symmetry of their Cayley tables verifies this. In contrast, the smallest non-abelian group, the dihedral group of order 6, does not have a symmetric Cayley table.
Associativity.
Because associativity is taken as an axiom when dealing with groups, it is often taken for granted when dealing with Cayley tables. However, Cayley tables can also be used to characterize the operation of a quasigroup, which does not assume associativity as an axiom (indeed, Cayley tables can be used to characterize the operation of any finite magma). Unfortunately, it is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table, as it is with commutativity. This is because associativity depends on a 3 term equation, $(ab)c=a(bc)$, while the Cayley table shows 2-term products. However, Light's associativity test can determine associativity with less effort than brute force.
Permutations.
Because the cancellation property holds for groups (and indeed even quasigroups), no row or column of a Cayley table may contain the same element twice. Thus each row and column of the table is a permutation of all the elements in the group. This greatly restricts which Cayley tables could conceivably define a valid group operation.
To see why a row or column cannot contain the same element more than once, let "a", "x", and "y" all be elements of a group, with "x" and "y" distinct. Then in the row representing the element "a", the column corresponding to "x" contains the product "ax", and similarly the column corresponding to "y" contains the product "ay". If these two products were equal – that is to say, row "a" contained the same element twice, our hypothesis – then "ax" would equal "ay". But because the cancellation law holds, we can conclude that if "ax" = "ay", then "x" = "y", a contradiction. Therefore, our hypothesis is incorrect, and a row cannot contain the same element twice. Exactly the same argument suffices to prove the column case, and so we conclude that each row and column contains no element more than once. Because the group is finite, the pigeonhole principle guarantees that each element of the group will be represented in each row and in each column exactly once.
Thus, the Cayley table of a group is an example of a latin square.
Another, maybe simpler proof: the cancellation property implies that for each x in the group, the one variable function of y f(x,y)= xy must be a one to one map. And one to one maps on finite sets are permutations.
Constructing Cayley tables.
Because of the structure of groups, one can very often "fill in" Cayley tables that have missing elements, even without having a full characterization of the group operation in question. For example, because each row and column must contain every element in the group, if all elements are accounted for save one, and there is one blank spot, without knowing anything else about the group it is possible to conclude that the element unaccounted for must occupy the remaining blank space. It turns out that this and other observations about groups in general allow us to construct the Cayley tables of groups knowing very little about the group in question. However, a Cayley table constructed using the method that follows may fail to meet the associativity requirement of a group, and therefore represent a quasigroup.
The "identity skeleton" of a finite group.
Inverses are identified by identity elements in the table. Because in any group, even a non-abelian group, every element commutes with its own inverse, it follows that the distribution of identity elements on the Cayley table will be symmetric across the table's diagonal. Those that lie on the diagonal are their own unique inverse.
Because the order of the rows and columns of a Cayley table is in fact arbitrary, it is convenient to order them in the following manner: beginning with the group's identity element, which is always its own inverse, list first all the elements that are their own inverse, followed by pairs of inverses listed adjacent to each other.
Then, for a finite group of a particular order, it is easy to characterize its "identity skeleton", so named because the identity elements on the Cayley table constructed in the manner described in the previous paragraph are clustered about the main diagonal – either they lie directly on it, or they are one removed from it.
It is relatively trivial to prove that groups with different identity skeletons cannot be isomorphic, though the converse is not true (for instance, the cyclic group "C8" and the quaternion group "Q" are non-isomorphic but have the same identity skeleton).
It is also the case that not all identity skeletons correspond to actual groups. For example, there is no six-element group with all elements their own inverses.
Permutation matrix generation.
The standard form of a Cayley table has the order of the elements in the rows the same as the order in the columns. Another form is to arrange the elements of the columns so that the "n"th column corresponds to the inverse of the element in the "n"th row. In our example of "D"3, we need only switch the last two columns, since "f" and "d" are the only elements that are not their own inverses, but instead inverses of each other.
This particular example lets us create six permutation matrices (all elements 1 or 0, exactly one 1 in each row and column). The 6x6 matrix representing an element will have a 1 in every position that has the letter of the element in the Cayley table and a zero in every other position, the Kronecker delta function for that symbol. (Note that "e" is in every position down the main diagonal, which gives us the identity matrix for 6x6 matrices in this case, as we would expect.) Here is the matrix that represents our element "a", for example.
This shows us directly that any group of order "n" is a subgroup of the permutation group "S""n", order "n"!.
Generalizations.
The above properties depend on some axioms valid for groups. It is natural to consider Cayley tables for other algebraic structures, such as for semigroups, quasigroups, and magmas, but some of the properties above do not hold.
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abstract_algebra
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In mathematics, the binary tetrahedral group, denoted 2T or ⟨2,3,3⟩, is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C. Shephard or 3[3]3 and by Coxeter, is isomorphic to the binary tetrahedral group.
The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin(3) ≅ Sp(1), where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Elements.
Explicitly, the binary tetrahedral group is given as the group of units in the ring of Hurwitz integers. There are 24 such units given by
$\{\pm 1,\pm i,\pm j,\pm k,\tfrac{1}{2}(\pm 1 \pm i \pm j \pm k)\}$
with all possible sign combinations.
All 24 units have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The convex hull of these 24 elements in 4-dimensional space form a convex regular 4-polytope called the 24-cell.
Properties.
The binary tetrahedral group, denoted by 2T, fits into the short exact sequence
$1\to\{\pm 1\}\to 2\mathrm{T}\to \mathrm{T} \to 1.$
This sequence does not split, meaning that 2T is "not" a semidirect product of {±1} by T. In fact, there is no subgroup of 2T isomorphic to T.
The binary tetrahedral group is the covering group of the tetrahedral group. Thinking of the tetrahedral group as the alternating group on four letters, T ≅ A4, we thus have the binary tetrahedral group as the covering group, 2T ≅ $\widehat{\mathrm{A}_4}$.
The center of 2T is the subgroup {±1}. The inner automorphism group is isomorphic to A4, and the full automorphism group is isomorphic to S4.
The binary tetrahedral group can be written as a semidirect product
$2\mathrm{T}=\mathrm{Q}\rtimes\mathrm{C}_3$
where Q is the quaternion group consisting of the 8 Lipschitz units and C3 is the cyclic group of order 3 generated by "ω" = −(1 + "i" + "j" + "k"). The group Z3 acts on the normal subgroup Q by conjugation. Conjugation by "ω" is the automorphism of Q that cyclically rotates i, j, and k.
One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) – the group of all 2 × 2 matrices over the finite field F3 with unit determinant, with this isomorphism covering the isomorphism of the projective special linear group PSL(2,3) with the alternating group A4.
Presentation.
The group 2T has a presentation given by
$\langle r,s,t \mid r^2 = s^3 = t^3 = rst \rangle$
or equivalently,
$\langle s,t \mid (st)^2 = s^3 = t^3 \rangle.$
Generators with these relations are given by
$r = i \qquad s = \tfrac{1}{2}(1+i+j+k) \qquad t = \tfrac{1}{2}(1+i+j-k),$
with $r^2 = s^3 = t^3 = -1$.
Subgroups.
The quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2T of index 3. This group and the center {±1} are the only nontrivial normal subgroups.
All other subgroups of 2T are cyclic groups generated by the various elements, with orders 3, 4, and 6.
Higher dimensions.
Just as the tetrahedral group generalizes to the rotational symmetry group of the "n"-simplex (as a subgroup of SO("n")), there is a corresponding higher binary group which is a 2-fold cover, coming from the cover Spin("n") → SO("n").
The rotational symmetry group of the "n"-simplex can be considered as the alternating group on "n" + 1 points, A"n"+1, and the corresponding binary group is a 2-fold covering group. For all higher dimensions except A6 and A7 (corresponding to the 5-dimensional and 6-dimensional simplexes), this binary group is the covering group (maximal cover) and is superperfect, but for dimensional 5 and 6 there is an additional exceptional 3-fold cover, and the binary groups are not superperfect.
Usage in theoretical physics.
The binary tetrahedral group was used in the context of Yang–Mills theory in 1956 by Chen Ning Yang and others.
It was first used in flavor physics model building by Paul Frampton and Thomas Kephart in 1994.
In 2012 it was shown that a relation between two neutrino mixing angles,
derived
by using this binary tetrahedral flavor symmetry, agrees with experiment.
Notes.
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abstract_algebra
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In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence
$0 \longrightarrow A \mathrel{\overset{q}{\longrightarrow}} B \mathrel{\overset{r}{\longrightarrow}} C \longrightarrow 0.$
If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to "split".
In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that:
"C" ≅ "B"/ker "r" ≅ "B"/"q"("A") (i.e., "C" isomorphic to the coimage of "r" or cokernel of "q")
to:
"B"
"q"("A") ⊕ "u"("C") ≅ "A" ⊕ "C"
where the first isomorphism theorem is then just the projection onto "C".
It is a categorical generalization of the rank–nullity theorem (in the form V ≅ ker "T" ⊕ im "T") in linear algebra.
Proof for the category of abelian groups.
3. ⇒ 1. and 3. ⇒ 2..
First, to show that 3. implies both 1. and 2., we assume 3. and take as "t" the natural projection of the direct sum onto "A", and take as "u" the natural injection of "C" into the direct sum.
1. ⇒ 3..
To prove that 1. implies 3., first note that any member of "B" is in the set (ker "t" + im "q"). This follows since for all "b" in "B", "b"
("b" − "qt"("b")) + "qt"("b"); "qt"("b") is in im "q", and "b" − "qt"("b") is in ker "t", since
"t"("b" − "qt"("b"))
"t"("b") − "tqt"("b")
"t"("b") − ("tq")"t"("b")
"t"("b") − "t"("b")
0.
Next, the intersection of im "q" and ker "t" is 0, since if there exists "a" in "A" such that "q"("a")
"b", and "t"("b")
0, then 0
"tq"("a")
"a"; and therefore, "b"
0.
This proves that "B" is the direct sum of im "q" and ker "t". So, for all "b" in "B", "b" can be uniquely identified by some "a" in "A", "k" in ker "t", such that "b"
"q"("a") + "k".
By exactness ker "r"
im "q". The subsequence "B" ⟶ "C" ⟶ 0 implies that "r" is onto; therefore for any "c" in "C" there exists some "b"
"q"("a") + "k" such that "c"
"r"("b")
"r"("q"("a") + "k")
"r"("k"). Therefore, for any "c" in "C", exists "k" in ker "t" such that "c" = "r"("k"), and "r"(ker "t") = "C".
If "r"("k")
0, then "k" is in im "q"; since the intersection of im "q" and ker "t"
0, then "k"
0. Therefore, the restriction "r": ker "t" → "C" is an isomorphism; and ker "t" is isomorphic to "C".
Finally, im "q" is isomorphic to "A" due to the exactness of 0 ⟶ "A" ⟶ "B"; so "B" is isomorphic to the direct sum of "A" and "C", which proves (3).
2. ⇒ 3..
To show that 2. implies 3., we follow a similar argument. Any member of "B" is in the set ker "r" + im "u"; since for all "b" in "B", "b"
("b" − "ur"("b")) + "ur"("b"), which is in ker "r" + im "u". The intersection of ker "r" and im "u" is 0, since if "r"("b")
0 and "u"("c")
"b", then 0
"ru"("c")
"c".
By exactness, im "q"
ker "r", and since "q" is an injection, im "q" is isomorphic to "A", so "A" is isomorphic to ker "r". Since "ru" is a bijection, "u" is an injection, and thus im "u" is isomorphic to "C". So "B" is again the direct sum of "A" and "C".
An alternative "abstract nonsense" proof of the splitting lemma may be formulated entirely in category theoretic terms.
Non-abelian groups.
In the form stated here, the splitting lemma does not hold in the full category of groups, which is not an abelian category.
Partially true.
It is partially true: if a short exact sequence of groups is left split or a direct sum (1. or 3.), then all of the conditions hold. For a direct sum this is clear, as one can inject from or project to the summands. For a left split sequence, the map "t" × "r": "B" → "A" × "C" gives an isomorphism, so "B" is a direct sum (3.), and thus inverting the isomorphism and composing with the natural injection "C" → "A" × "C" gives an injection "C" → "B" splitting "r" (2.).
However, if a short exact sequence of groups is right split (2.), then it need not be left split or a direct sum (neither 1. nor 3. follows): the problem is that the image of the right splitting need not be normal. What is true in this case is that "B" is a semidirect product, though not in general a direct product.
Counterexample.
To form a counterexample, take the smallest non-abelian group "B" ≅ "S"3, the symmetric group on three letters. Let "A" denote the alternating subgroup, and let "C"
"B"/"A" ≅ {±1}. Let "q" and "r" denote the inclusion map and the sign map respectively, so that
$0 \longrightarrow A \mathrel{\stackrel{q}{\longrightarrow}} B \mathrel{\stackrel{r}{\longrightarrow}} C \longrightarrow 0 $
is a short exact sequence. 3. fails, because "S"3 is not abelian, but 2. holds: we may define "u": "C" → "B" by mapping the generator to any two-cycle. Note for completeness that 1. fails: any map "t": "B" → "A" must map every two-cycle to the identity because the map has to be a group homomorphism, while the order of a two-cycle is 2 which can not be divided by the order of the elements in "A" other than the identity element, which is 3 as "A" is the alternating subgroup of "S"3, or namely the cyclic group of order 3. But every permutation is a product of two-cycles, so "t" is the trivial map, whence "tq": "A" → "A" is the trivial map, not the identity.
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abstract_algebra
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Algebraic-group factorisation algorithms are algorithms for factoring an integer "N" by working in an algebraic group defined modulo "N" whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors "p"1, "p"2, ... By the Chinese remainder theorem, arithmetic modulo "N" corresponds to arithmetic in all the reduced groups simultaneously.
The aim is to find an element which is not the identity of the group modulo "N", but is the identity modulo one of the factors, so a method for recognising such "one-sided identities" is required. In general, one finds them by performing operations that move elements around and leave the identities in the reduced groups unchanged. Once the algorithm finds a one-sided identity all future terms will also be one-sided identities, so checking periodically suffices.
Computation proceeds by picking an arbitrary element "x" of the group modulo "N" and computing a large and smooth multiple "Ax" of it; if the order of at least one but not all of the reduced groups is a divisor of A, this yields a factorisation. It need not be a prime factorisation, as the element might be an identity in more than one of the reduced groups.
Generally, A is taken as a product of the primes below some limit K, and "Ax" is computed by successive multiplication of "x" by these primes; after each multiplication, or every few multiplications, the check is made for a one-sided identity.
The two-step procedure.
It is often possible to multiply a group element by several small integers more quickly than by their product, generally by difference-based methods; one calculates differences between consecutive primes and adds consecutively by the $d_i r$. This means that a two-step procedure becomes sensible, first computing "Ax" by multiplying "x" by all the primes below a limit B1, and then examining "p Ax" for all the primes between B1 and a larger limit B2.
Methods corresponding to particular algebraic groups.
If the algebraic group is the multiplicative group mod "N", the one-sided identities are recognised by computing greatest common divisors with "N", and the result is the "p" − 1 method.
If the algebraic group is the multiplicative group of a quadratic extension of "N", the result is the "p" + 1 method; the calculation involves pairs of numbers modulo "N". It is not possible to tell whether $\mathbb Z/N\mathbb Z [ \sqrt t]$ is actually a quadratic extension of $\mathbb Z/N\mathbb Z $ without knowing the factorisation of "N". This requires knowing whether "t" is a quadratic residue modulo "N", and there are no known methods for doing this without knowledge of the factorisation. However, provided "N" does not have a very large number of factors, in which case another method should be used first, picking random "t" (or rather picking "A" with "t" = "A"2 − 4) will accidentally hit a quadratic non-residue fairly quickly. If "t" is a quadratic residue, the p+1 method degenerates to a slower form of the "p" − 1 method.
If the algebraic group is an elliptic curve, the one-sided identities can be recognised by failure of inversion in the elliptic-curve point addition procedure, and the result is the elliptic curve method; Hasse's theorem states that the number of points on an elliptic curve modulo "p" is always within $2 \sqrt p$ of "p".
All three of the above algebraic groups are used by the GMP-ECM package, which includes efficient implementations of the two-stage procedure, and an implementation of the PRAC group-exponentiation algorithm which is rather more efficient than the standard binary exponentiation approach.
The use of other algebraic groups—higher-order extensions of "N" or groups corresponding to algebraic curves of higher genus—is occasionally proposed, but almost always impractical. These methods end up with smoothness constraints on numbers of the order of "p""d" for some "d" > 1, which are much less likely to be smooth than numbers of the order of "p".
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1902874
|
abstract_algebra
|
Type of classification in algebra
In abstract algebra, a branch of mathematics, an Archimedean group is a linearly ordered group for which the Archimedean property holds: every two positive group elements are bounded by integer multiples of each other. The set R of real numbers together with the operation of addition and the usual ordering relation between pairs of numbers is an Archimedean group. By a result of Otto Hölder, every Archimedean group is isomorphic to a subgroup of this group. The name "Archimedean" comes from Otto Stolz, who named the Archimedean property after its appearance in the works of Archimedes.
Definition.
An additive group consists of a set of elements, an associative addition operation that combines pairs of elements and returns a single element,
an identity element (or zero element) whose sum with any other element is the other element, and an additive inverse operation such that the sum of any element and its inverse is zero.
A group is a linearly ordered group when, in addition, its elements can be linearly ordered in a way that is compatible with the group operation: for all elements "x", "y", and "z", if "x" ≤ "y" then "x" + "z" ≤ "y" + "z" and "z" + "x" ≤ "z" + "y".
The notation "na" (where "n" is a natural number) stands for the group sum of "n" copies of "a".
An Archimedean group ("G", +, ≤) is a linearly ordered group subject to the following additional condition, the Archimedean property: For every "a" and "b" in "G" which are greater than 0, it is possible to find a natural number "n" for which the inequality "b" ≤ "na" holds.
An equivalent definition is that an Archimedean group is a linearly ordered group without any bounded cyclic subgroups: there does not exist a cyclic subgroup "S" and an element "x" with "x" greater than all elements in "S". It is straightforward to see that this is equivalent to the other definition: the Archimedean property for a pair of elements "a" and "b" is just the statement that the cyclic subgroup generated by "a" is not bounded by "b".
Examples of Archimedean groups.
The sets of the integers, the rational numbers, and the real numbers, together with the operation of addition and the usual ordering (≤), are Archimedean groups. Every subgroup of an Archimedean group is itself Archimedean, so it follows that every subgroup of these groups, such as the additive group of the even numbers or of the dyadic rationals, also forms an Archimedean group.
Conversely, as Otto Hölder showed, every Archimedean group is isomorphic (as an ordered group) to a subgroup of the real numbers. It follows from this that every Archimedean group is necessarily an abelian group: its addition operation must be commutative.
Examples of non-Archimedean groups.
Groups that cannot be linearly ordered, such as the finite groups, are not Archimedean. For another example, see the "p"-adic numbers, a system of numbers generalizing the rational numbers in a different way to the real numbers.
Non-Archimedean ordered groups also exist; the ordered group ("G", +, ≤) defined as follows is not Archimedean. Let the elements of "G" be the points of the Euclidean plane, given by their Cartesian coordinates: pairs ("x", "y") of real numbers. Let the group addition operation be pointwise (vector) addition, and order these points in lexicographic order: if "a" = ("u", "v") and "b" = ("x", "y"), then "a" + "b" = ("u" + "x", "v" + "y"), and
"a" ≤ "b" exactly when either "v" < "y" or "v" = "y" and "u" ≤ "x". Then this gives an ordered group, but one that is not Archimedean. To see this, consider the elements (1, 0) and (0, 1), both of which are greater than the zero element of the group (the origin). For every natural number "n", it follows from these definitions that "n" (1, 0) = ("n", 0) < (0, 1), so there is no "n" that satisfies the Archimedean property. This group can be thought of as the additive group of pairs of a real number and an infinitesimal, $(x, y) = x \epsilon + y,$ where $\epsilon$ is a unit infinitesimal: $\epsilon > 0$ but $\epsilon < y$ for any positive real number $y > 0$. Non-Archimedean ordered fields can be defined similarly, and their additive groups are non-Archimedean ordered groups. These are used in non-standard analysis, and include the hyperreal numbers and surreal numbers.
While non-Archimedean ordered groups cannot be embedded in the real numbers, they can be embedded in a power of the real numbers, with lexicographic order, by the Hahn embedding theorem; the example above is the 2-dimensional case.
Additional properties.
Every Archimedean group has the property that, for every Dedekind cut of the group, and every group element ε > 0, there exists another group element "x" with "x" on the lower side of the cut and "x" + ε on the upper side of the cut. However, there exist non-Archimedean ordered groups with the same property. The fact that Archimedean groups are abelian can be generalized: every ordered group with this property is abelian.
Generalisations.
Archimedean groups can be generalised to Archimedean monoids, linearly ordered monoids that obey the Archimedean property. Examples include the natural numbers, the non-negative rational numbers, and the non-negative real numbers, with the usual binary operation $+$ and order $«/math>. Through a similar proof as for Archimedean groups, Archimedean monoids can be shown to be commutative.
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128698
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abstract_algebra
|
In mathematics, the binary octahedral group, name as 2O or ⟨2,3,4⟩ is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group "O" or (2,3,4) of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism $\operatorname{Spin}(3) \to \operatorname{SO}(3)$ of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48.
The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism $\operatorname{Spin}(3) \cong \operatorname{Sp}(1)$ where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Elements.
Explicitly, the binary octahedral group is given as the union of the 24 Hurwitz units
$\{\pm 1,\pm i,\pm j,\pm k,\tfrac{1}{2}(\pm 1 \pm i \pm j \pm k)\}$
with all 24 quaternions obtained from
$\tfrac{1}{\sqrt 2}(\pm 1 \pm 1i + 0j + 0k)$
by a permutation of coordinates and all possible sign combinations. All 48 elements have absolute value 1 and therefore lie in the unit quaternion group Sp(1).
Properties.
The binary octahedral group, denoted by 2"O", fits into the short exact sequence
$1\to\{\pm 1\}\to 2O\to O \to 1.\,$
This sequence does not split, meaning that 2"O" is "not" a semidirect product of {±1} by "O". In fact, there is no subgroup of 2"O" isomorphic to "O".
The center of 2"O" is the subgroup {±1}, so that the inner automorphism group is isomorphic to "O". The full automorphism group is isomorphic to "O" × Z2.
Presentation.
The group 2"O" has a presentation given by
$\langle r,s,t \mid r^2 = s^3 = t^4 = rst \rangle$
or equivalently,
$\langle s,t \mid (st)^2 = s^3 = t^4 \rangle.$
Quaternion generators with these relations are given by
$r = \tfrac{1}{\sqrt 2}(i+j) \qquad s = \tfrac{1}{2}(1+i+j+k) \qquad t = \tfrac{1}{\sqrt 2}(1+i),$
with $ r^2 = s^3 = t^4 = rst = -1.$
Subgroups.
The binary tetrahedral group, 2"T", consisting of the 24 Hurwitz units, forms a normal subgroup of index 2. The quaternion group, "Q"8, consisting of the 8 Lipschitz units forms a normal subgroup of 2"O" of index 6. The quotient group is isomorphic to "S"3 (the symmetric group on 3 letters). These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2"O".
The generalized quaternion group, "Q"16, also forms a subgroup of 2"O", index 3. This subgroup is self-normalizing so its conjugacy class has 3 members. There are also isomorphic copies of the binary dihedral groups "Q"8 and "Q"12 in 2"O".
All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).
Higher dimensions.
The "binary octahedral group" generalizes to higher dimensions: just as the octahedron generalizes to the orthoplex, the octahedral group in SO(3) generalizes to the hyperoctahedral group in SO("n"), which has a binary cover under the map $\operatorname{Spin}(n) \to SO(n).$
Notes.
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1557510
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abstract_algebra
|
Mathematical abelian group
In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one.
It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation),
as the group of bitwise exclusive or operations on two-bit binary values,
or more abstractly as Z2 × Z2, the direct product of two copies of the cyclic group of order 2.
It was named Vierergruppe (meaning four-group) by Felix Klein in 1884.
It is also called the Klein group, and is often symbolized by the letter V or as K4.
The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups.
Presentations.
The Klein group's Cayley table is given by:
The Klein four-group is also defined by the group presentation
$\mathrm{V} = \left\langle a,b \mid a^2 = b^2 = (ab)^2 = e \right\rangle.$
All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation. The Klein four-group is the smallest non-cyclic group. It is however an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, i.e. D4 (or D2, using the geometric convention); other than the group of order 2, it is the only dihedral group that is abelian.
The Klein four-group is also isomorphic to the direct sum Z2 ⊕ Z2, so that it can be represented as the pairs {(0,0), (0,1), (1,0), (1,1)} under component-wise addition modulo 2 (or equivalently the bit strings {00, 01, 10, 11}under bitwise XOR); with (0,0) being the group's identity element. The Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. The Klein four-group is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements, i.e. over a field of sets with four elements, e.g. $\{\emptyset, \{\alpha\}, \{\beta\}, \{\alpha, \beta\}\}$; the empty set is the group's identity element in this case.
Another numerical construction of the Klein four-group is the set { 1, 3, 5, 7 }, with the operation being multiplication modulo 8. Here "a" is 3, "b" is 5, and "c" = "ab" is 3 × 5 = 15 ≡ 7 (mod 8).
The Klein four-group has a representation as 2×2 real matrices with the operation being matrix multiplication:
$
1 & 0\\
0 & 1
\end{pmatrix},\quad
1 & 0\\
0 & -1
\end{pmatrix},\quad
-1 & 0\\
0 & 1
\end{pmatrix},\quad
-1 & 0\\
0 & -1
$
On a Rubik's Cube the "4 dots" pattern can be made in three ways, depending on the pair of faces that are left blank; these three positions together with the "identity" or home position form an example of the Klein group.
Geometry.
Geometrically, in two dimensions the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.
In three dimensions there are three different symmetry groups that are algebraically the Klein four-group V:
Permutation representation.
The three elements of order two in the Klein four-group are interchangeable: the automorphism group of V is the group of permutations of these three elements, that is, S3.
The Klein four-group's permutations of its own elements can be thought of abstractly as its permutation representation on four points:
V = { (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }
In this representation, V is a normal subgroup of the alternating group A4
(and also the symmetric group S4) on four letters. In fact, it is the kernel of a surjective group homomorphism from S4 to S3.
Other representations within S4 are:
{ (), (1,2), (3,4), (1,2)(3,4) }
{ (), (1,3), (2,4), (1,3)(2,4) }
{ (), (1,4), (2,3), (1,4)(2,3) }
They are not normal subgroups of S4.
Algebra.
According to Galois theory, the existence of the Klein four-group (and in particular, the permutation representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals, as established by Lodovico Ferrari:
the map S4 → S3 corresponds to the resolvent cubic, in terms of Lagrange resolvents.
In the construction of finite rings, eight of the eleven rings with four elements have the Klein four-group as their additive substructure.
If R× denotes the multiplicative group of non-zero reals and R+ the multiplicative group of positive reals, R× × R× is the group of units of the ring R × R, and R+ × R+ is a subgroup of R× × R× (in fact it is the component of the identity of R× × R×). The quotient group (R× × R×) / (R+ × R+) is isomorphic to the Klein four-group. In a similar fashion, the group of units of the split-complex number ring, when divided by its identity component, also results in the Klein four-group.
Graph theory.
The simplest simple connected graph that admits the Klein four-group as its automorphism group is the diamond graph shown below. It is also the automorphism group of some other graphs that are simpler in the sense of having fewer entities. These include the graph with four vertices and one edge, which remains simple but loses connectivity, and the graph with two vertices connected to each other by two edges, which remains connected but loses simplicity.
Music.
In music composition the four-group is the basic group of permutations in the twelve-tone technique. In that instance the Cayley table is written;
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21489
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abstract_algebra
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Every subgroup of a cyclic group is cyclic, and if finite, its order divides its parent's
In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order "n", every subgroup's order is a divisor of "n", and there is exactly one subgroup for each divisor. This result has been called the fundamental theorem of cyclic groups.
Finite cyclic groups.
For every finite group "G" of order "n", the following statements are equivalent:
If either (and thus both) are true, it follows that there exists exactly one subgroup of order "d", for any divisor of "n".
This statement is known by various names such as characterization by subgroups. (See also cyclic group for some characterization.)
There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the Klein group is an example. However, the Klein group has more than one subgroup of order 2, so it does not meet the conditions of the characterization.
The infinite cyclic group.
The infinite cyclic group is isomorphic to the additive subgroup Z of the integers. There is one subgroup "d"Z for each integer "d" (consisting of the multiples of "d"), and with the exception of the trivial group (generated by "d" = 0) every such subgroup is itself an infinite cyclic group. Because the infinite cyclic group is a free group on one generator (and the trivial group is a free group on no generators), this result can be seen as a special case of the Nielsen–Schreier theorem that every subgroup of a free group is itself free.
The fundamental theorem for finite cyclic groups can be established from the same theorem for the infinite cyclic groups, by viewing each finite cyclic group as a quotient group of the infinite cyclic group.
Lattice of subgroups.
In both the finite and the infinite case, the lattice of subgroups of a cyclic group is isomorphic to the dual of a divisibility lattice. In the finite case, the lattice of subgroups of a cyclic group of order "n" is isomorphic to the dual of the lattice of divisors of "n", with a subgroup of order "n"/"d" for each divisor "d". The subgroup of order "n"/"d" is a subgroup of the subgroup of order "n"/"e" if and only if "e" is a divisor of "d". The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers, then the subgroup generated by "d" is a subgroup of the subgroup generated by "e" if and only if "e" is a divisor of "d".
Divisibility lattices are distributive lattices, and therefore so are the lattices of subgroups of cyclic groups. This provides another alternative characterization of the finite cyclic groups: they are exactly the finite groups whose lattices of subgroups are distributive. More generally, a finitely generated group is cyclic if and only if its lattice of subgroups is distributive and an arbitrary group is locally cyclic if and only its lattice of subgroups is distributive. The additive group of the rational numbers provides an example of a group that is locally cyclic, and that has a distributive lattice of subgroups, but that is not itself cyclic.
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1541864
|
abstract_algebra
|
In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup "A" acts transitively on certain subgroups normalized by "A". It originated in the proof of the odd order theorem by Feit and Thompson (1963), where it was used to prove the Thompson uniqueness theorem.
Statement.
Suppose that "G" is a finite group and "p" a prime such that all "p"-local subgroups are "p"-constrained. If "A" is a self-centralizing normal abelian subgroup of a "p"-Sylow subgroup such that "A" has rank at least 3, then the centralizer C"G"("A") act transitively on the maximal "A"-invariant "q" subgroups of "G" for any prime "q" ≠ "p".
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3761379
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abstract_algebra
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In mathematics, the Walter theorem, proved by John H. Walter (1967, 1969), describes the finite groups whose Sylow 2-subgroup is abelian. used Bender's method to give a simpler proof.
Statement.
Walter's theorem states that if "G" is a finite group whose 2-sylow subgroups are abelian, then "G"/"O"("G") has a normal subgroup of odd index that is a product of groups each of which is a 2-group or one of the simple groups PSL2("q") for "q" = 2"n" or "q" = 3 or 5 mod 8, or the Janko group J1, or Ree groups 2"G"2(32"n"+1). (Here "O(G)" denotes the unique largest normal subgroup of "G" of
odd order.)
The original statement of Walter's theorem did not quite identify the Ree groups, but only stated that the corresponding groups have a similar subgroup structure as Ree groups. Thompson (1967, 1972, 1977) and later showed that they are all Ree groups, and gave a unified exposition of this result.
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3256934
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abstract_algebra
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Mathematic group theory
In mathematics, Burnside's theorem in group theory states that if "G" is a finite group of order $p^a q^b$ where "p" and "q" are prime numbers, and "a" and "b" are non-negative integers, then "G" is solvable. Hence each
non-Abelian finite simple group has order divisible by at least three distinct primes.
History.
The theorem was proved by William Burnside (1904) using the representation theory of finite groups. Several special cases of the theorem had previously been proved by Burnside, Jordan, and Frobenius. John Thompson pointed out that a proof avoiding the use of representation theory could be extracted from his work on the N-group theorem, and this was done explicitly by for groups of odd order, and by for groups of even order. simplified the proofs.
Proof.
The following proof — using more background than Burnside's — is by contradiction. Let "paqb" be the smallest product of two prime powers, such that there is a non-solvable group "G" whose order is equal to this number.
*"G" is a simple group with trivial center and "a" is not zero.
If "G" had a nontrivial proper normal subgroup "H", then (because of the minimality of "G"), "H" and "G"/"H" would be solvable, so "G" as well, which would contradict our assumption. So "G" is simple.
If "a" were zero, "G" would be a finite q-group, hence nilpotent, and therefore solvable.
Similarly, "G" cannot be abelian, otherwise it would be solvable. As "G" is simple, its center must therefore be trivial.
* There is an element "g" of "G" which has "qd" conjugates, for some "d" > 0.
By the first statement of Sylow's theorem, "G" has a subgroup "S" of order "pa". Because "S" is a nontrivial "p"-group, its center "Z"("S") is nontrivial. Fix a nontrivial element $g\in Z(S)$. The number of conjugates of "g" is equal to the index of its stabilizer subgroup "Gg", which divides the index "qb" of "S" (because "S" is a subgroup of "Gg"). Hence this number is of the form "qd". Moreover, the integer "d" is strictly positive, since "g" is nontrivial and therefore not central in "G".
* There exists a nontrivial irreducible representation ρ with character χ, such that its dimension "n" is not divisible by "q" and the complex number "χ"("g") is not zero.
Let ("χ""i")1 ≤ "i" ≤ "h" be the family of irreducible characters of "G" over $\mathbb{C}$ (here "χ"1 denotes the trivial character). Because "g" is not in the same conjugacy class as 1, the orthogonality relation for the columns of the group's character table gives:
$0=\sum_{i=1}^h \chi_i(1)\chi_i(g)= 1 + \sum_{i=2}^h \chi_i(1)\chi_i(g).$
Now the "χ""i"("g") are algebraic integers, because they are sums of roots of unity. If all the nontrivial irreducible characters which don't vanish at "g" take a value divisible by "q" at 1, we deduce that
$-\frac1q=\sum_{i\ge 2,~\chi_i(g)\ne 0}\frac{\chi_i(1)}q\chi_i(g)$
is an algebraic integer (since it is a sum of integer multiples of algebraic integers), which is absurd. This proves the statement.
* The complex number "q""d""χ"("g")/"n" is an algebraic integer.
The set of integer-valued class functions on "G", "Z"($\mathbb{Z}$["G"]), is a commutative ring, finitely generated over $\mathbb{Z}$. All of its elements are thus integral over $\mathbb{Z}$, in particular the mapping "u" which takes the value 1 on the conjugacy class of g and 0 elsewhere.
The mapping $A:Z(\mathbb{Z}[G]) \rightarrow \operatorname{End}(\mathbb{C}^n)$ which sends a class function "f" to
$\sum_{s\in G} f(s)\rho(s)$
is a ring homomorphism. Because "ρ"("s")−1"A"("u")"ρ"("s") = "A"("u") for all "s", Schur's lemma implies that "A"("u") is a homothety λIn. Its trace "nλ" is equal to
$\sum_{s\in G} f(s)\chi(s)=q^d\chi(g).$
Because the homothety "λI""n" is the homomorphic image of an integral element, this proves that the complex number "λ" = "q""d""χ"("g")/"n" is an algebraic integer.
* The complex number "χ"("g")/"n" is an algebraic integer.
Since "q" is relatively prime to "n", by Bézout's identity there are two integers "x" and "y" such that:
$xq^d + yn=1\quad\text{therefore}\quad \frac{\chi(g)}{n}=x\frac{q^d\chi(g)}{n} + y\chi(g).$
Because a linear combination with integer coefficients of algebraic integers is again an algebraic integer, this proves the statement.
* The image of "g", under the representation "ρ", is a homothety.
Let "ζ" be the complex number "χ"("g")/"n". It is an algebraic integer, so its norm "N"("ζ") (i.e. the product of its conjugates, that is the roots of its minimal polynomial over $\mathbb{Q}$) is a nonzero integer. Now "ζ" is the average of roots of unity (the eigenvalues of "ρ"("g")), hence so are its conjugates, so they all have an absolute value less than or equal to 1. Because the absolute value of their product "N"("ζ") is greater than or equal to 1, their absolute value must all be 1, in particular "ζ", which means that the eigenvalues of "ρ"("g") are all equal, so "ρ"("g") is a homothety.
* Conclusion
Let "N" be the kernel of "ρ". The homothety "ρ"("g") is central in Im("ρ") (which is canonically isomorphic to "G"/"N"), whereas "g" is not central in "G". Consequently, the normal subgroup "N" of the simple group "G" is nontrivial, hence it is equal to "G", which contradicts the fact that ρ is a nontrivial representation.
This contradiction proves the theorem.
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512355
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abstract_algebra
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Group that has an upper central series terminating with G
In mathematics, specifically group theory, a nilpotent group "G" is a group that has an upper central series that terminates with "G". Equivalently, its central series is of finite length or its lower central series terminates with {1}.
Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.
Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.
Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series.
Definition.
The definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group G:
For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G; and G is said to be nilpotent of class n. (By definition, the length is n if there are $n + 1$ different subgroups in the series, including the trivial subgroup and the whole group.)
Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series.
If a group has nilpotency class at most n, then it is sometimes called a nil-n group.
It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups.
Examples.
The natural numbers "k" for which any group of order "k" is nilpotent have been characterized (sequence in the OEIS).
Explanation of term.
Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group $G$ of nilpotence degree $n$ and an element $g$, the function $\operatorname{ad}_g \colon G \to G$ defined by $\operatorname{ad}_g(x) := [g,x]$ (where $[g,x]=g^{-1} x^{-1} g x$ is the commutator of $g$ and $x$) is nilpotent in the sense that the $n$th iteration of the function is trivial: $\left(\operatorname{ad}_g\right)^n(x)=e$ for all $x$ in $G$.
This is not a defining characteristic of nilpotent groups: groups for which $\operatorname{ad}_g$ is nilpotent of degree $n$ (in the sense above) are called $n$-Engel groups, and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated.
An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).
Properties.
Since each successive factor group "Z""i"+1/"Z""i" in the upper central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.
Every subgroup of a nilpotent group of class "n" is nilpotent of class at most "n"; in addition, if "f" is a homomorphism of a nilpotent group of class "n", then the image of "f" is nilpotent of class at most "n".
The following statements are equivalent for finite groups, revealing some useful properties of nilpotency:
Proof:
Statement (d) can be extended to infinite groups: if "G" is a nilpotent group, then every Sylow subgroup "G""p" of "G" is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in "G" (see torsion subgroup).
Many properties of nilpotent groups are shared by hypercentral groups.
Notes.
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In mathematics, specifically in group theory, the direct product is an operation that takes two groups "G" and "H" and constructs a new group, usually denoted "G" × "H". This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted $G \oplus H$. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
Definition.
Given groups "G "(with operation *) and "H" (with operation ∆), the direct product "G" × "H" is defined as follows:
The resulting algebraic object satisfies the axioms for a group. Specifically:
("x"1, "y"1) + ("x"2, "y"2)
Examples.
("x"1 + "x"2, "y"1 + "y"2).
("x"1, "y"1) × ("x"2, "y"2)
("x"1 × "x"2, "y"1 × "y"2).
Then the direct product "G" × "H" is isomorphic to the Klein four-group:
Algebraic structure.
Let "G" and "H" be groups, let "P" = "G" × "H", and consider the following two subsets of "P":
"G"′ = { ("g", 1) : "g" ∈ "G" } and "H"′ = { (1, "h") : "h" ∈ "H" }.
Both of these are in fact subgroups of "P", the first being isomorphic to "G", and the second being isomorphic to "H". If we identify these with "G" and "H", respectively, then we can think of the direct product "P" as containing the original groups "G" and "H" as subgroups.
These subgroups of "P" have the following three important properties:
Together, these three properties completely determine the algebraic structure of the direct product "P". That is, if "P" is "any" group having subgroups "G" and "H" that satisfy the properties above, then "P" is necessarily isomorphic to the direct product of "G" and "H". In this situation, "P" is sometimes referred to as the internal direct product of its subgroups "G" and "H".
In some contexts, the third property above is replaced by the following:
3′. Both "G" and "H" are normal in "P".
This property is equivalent to property 3, since the elements of two normal subgroups with trivial intersection necessarily commute, a fact which can be deduced by considering the commutator ["g","h"] of any "g" in "G", "h" in "H".
Presentations.
The algebraic structure of "G" × "H" can be used to give a presentation for the direct product in terms of the presentations of "G" and "H". Specifically, suppose that
$G = \langle S_G \mid R_G \rangle \ \ $ and $\ \ H = \langle S_H \mid R_H \rangle,$
where $S_G$ and $S_H$ are (disjoint) generating sets and $R_G$ and $R_H$ are defining relations. Then
$G \times H = \langle S_G \cup S_H \mid R_G \cup R_H \cup R_P \rangle$
where $R_P$ is a set of relations specifying that each element of $S_G$ commutes with each element of $S_H$.
For example if
$G = \langle a \mid a^3=1 \rangle \ \ $ and $\ \ H = \langle b \mid b^5=1 \rangle$
then
$G \times H = \langle a, b \mid a^3 = 1, b^5 = 1, ab=ba \rangle.$
Normal structure.
As mentioned above, the subgroups "G" and "H" are normal in "G" × "H". Specifically, define functions "πG": "G" × "H" → "G" and "πH": "G" × "H" → "H" by
"πG"("g", "h") = "g" and "πH"("g", "h") = "h".
Then "πG" and "πH" are homomorphisms, known as projection homomorphisms, whose kernels are "H" and "G", respectively.
It follows that "G" × "H" is an extension of "G" by "H" (or vice versa). In the case where "G" × "H" is a finite group, it follows that the composition factors of "G" × "H" are precisely the union of the composition factors of "G" and the composition factors of "H".
Further properties.
Universal property.
The direct product "G" × "H" can be characterized by the following universal property. Let "πG": "G" × "H" → "G" and "πH": "G" × "H" → "H" be the projection homomorphisms. Then for any group "P" and any homomorphisms ƒ"G": "P" → "G" and ƒ"H": "P" → "H", there exists a unique homomorphism ƒ: "P" → "G" × "H" making the following diagram commute:
Specifically, the homomorphism ƒ is given by the formula
ƒ("p")
( ƒ"G"("p"), ƒ"H"("p") ).
This is a special case of the universal property for products in category theory.
Subgroups.
If "A" is a subgroup of "G" and "B" is a subgroup of "H", then the direct product "A" × "B" is a subgroup of "G" × "H". For example, the isomorphic copy of "G" in "G" × "H" is the product "G" × {1}, where is the trivial subgroup of "H".
If "A" and "B" are normal, then "A" × "B" is a normal subgroup of "G" × "H". Moreover, the quotient of the direct products is isomorphic to the direct product of the quotients:
("G" × "H") / ("A" × "B") ≅ ("G" / "A") × ("H" / "B").
Note that it is not true in general that every subgroup of "G" × "H" is the product of a subgroup of "G" with a subgroup of "H". For example, if "G" is any non-trivial group, then the product "G" × "G" has a diagonal subgroup
Δ
{ ("g", "g") : "g" ∈ "G" }
which is not the direct product of two subgroups of "G".
The subgroups of direct products are described by Goursat's lemma. Other subgroups include fiber products of "G" and "H".
Conjugacy and centralizers.
Two elements ("g"1, "h"1) and ("g"2, "h"2) are conjugate in "G" × "H" if and only if "g"1 and "g"2 are conjugate in "G" and "h"1 and "h"2 are conjugate in "H". It follows that each conjugacy class in "G" × "H" is simply the Cartesian product of a conjugacy class in "G" and a conjugacy class in "H".
Along the same lines, if ("g", "h") ∈ "G" × "H", the centralizer of ("g", "h") is simply the product of the centralizers of "g" and "h":
"C""G"×"H"("g", "h") = "C""G"("g") × "C""H"("h").
Similarly, the center of "G" × "H" is the product of the centers of "G" and "H":
"Z"("G" × "H") = "Z"("G") × "Z"("H").
Normalizers behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.
Automorphisms and endomorphisms.
If "α" is an automorphism of "G" and "β" is an automorphism of "H", then the product function "α" × "β": "G" × "H" → "G" × "H" defined by
("α" × "β")("g", "h")
("α"("g"), "β"("h"))
is an automorphism of "G" × "H". It follows that Aut("G" × "H") has a subgroup isomorphic
to the direct product Aut("G") × Aut("H").
It is not true in general that every automorphism of "G" × "H" has the above form. (That is, Aut("G") × Aut("H") is often a proper subgroup of Aut("G" × "H").) For example, if "G" is any group, then there exists an automorphism "σ" of "G" × "G" that switches the two factors, i.e.
"σ"("g"1, "g"2)
("g"2, "g"1).
For another example, the automorphism group of Z × Z is "GL"(2, Z), the group of all 2 × 2 matrices with integer entries and determinant, ±1. This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.
In general, every endomorphism of "G" × "H" can be written as a 2 × 2 matrix
$\begin{bmatrix}\alpha & \beta \\ \gamma & \delta\end{bmatrix}$
where "α" is an endomorphism of "G", "δ" is an endomorphism of "H", and "β": "H" → "G" and "γ": "G" → "H" are homomorphisms. Such a matrix must have the property that every element in the image of "α" commutes with every element in the image of "β", and every element in the image of "γ" commutes with every element in the image of "δ".
When "G" and "H" are indecomposable, centerless groups, then the automorphism group is relatively straightforward, being Aut("G") × Aut("H") if "G" and "H" are not isomorphic, and Aut("G") wr 2 if "G" ≅ "H", wr denotes the wreath product. This is part of the Krull–Schmidt theorem, and holds more generally for finite direct products.
Generalizations.
Finite direct products.
It is possible to take the direct product of more than two groups at once. Given a finite sequence "G"1, ..., "G""n" of groups, the direct product
$\prod_{i=1}^n G_i \;=\; G_1 \times G_2 \times \cdots \times G_n$
is defined as follows:
This has many of the same properties as the direct product of two groups, and can be characterized algebraically in a similar way.
Infinite direct products.
It is also possible to take the direct product of an infinite number of groups. For an infinite sequence "G"1, "G"2, ... of groups, this can be defined just like the finite direct product of above, with elements of the infinite direct product being infinite tuples.
More generally, given an indexed family { "Gi" }"i"∈"I" of groups, the direct product Π"i"∈"I" "Gi" is defined as follows:
Unlike a finite direct product, the infinite direct product Π"i"∈"I" "Gi" is not generated by the elements of the isomorphic subgroups { "Gi" }"i"∈"I". Instead, these subgroups generate a subgroup of the direct product known as the infinite direct sum, which consists of all elements that have only finitely many non-identity components.
Other products.
Semidirect products.
Recall that a group "P" with subgroups "G" and "H" is isomorphic to the direct product of "G" and "H" as long as it satisfies the following three conditions:
A semidirect product of "G" and "H" is obtained by relaxing the third condition, so that only one of the two subgroups "G", "H" is required to be normal. The resulting product still consists of ordered pairs ("g", "h"), but with a slightly more complicated rule for multiplication.
It is also possible to relax the third condition entirely, requiring neither of the two subgroups to be normal. In this case, the group "P" is referred to as a Zappa–Szép product of "G" and "H".
Free products.
The free product of "G" and "H", usually denoted "G" ∗ "H", is similar to the direct product, except that the subgroups "G" and "H" of "G" ∗ "H" are not required to commute. That is, if
"G" = 〈 "SG"|"RG" 〉 and "H" = 〈 "SH"|"RH" 〉,
are presentations for "G" and "H", then
"G" ∗ "H" = 〈 "SG" ∪ "SH"|"RG" ∪ "RH" 〉.
Unlike the direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The free product is actually the coproduct in the category of groups.
Subdirect products.
If "G" and "H" are groups, a subdirect product of "G" and "H" is any subgroup of "G" × "H" which maps surjectively onto "G" and "H" under the projection homomorphisms. By Goursat's lemma, every subdirect product is a fiber product.
Fiber products.
Let "G", "H", and "Q" be groups, and let "φ": "G" → "Q" and "χ": "H" → "Q" be homomorphisms. The fiber product of "G" and "H" over "Q", also known as a pullback, is the following subgroup of "G" × "H":
<math display=block>G \times_{Q} H = \{ \, (g, h) \in G \times H : \phi(g) = \chi(h) \,\}\text{.}$
If "φ": "G" → "Q" and "χ": "H" → "Q" are epimorphisms, then this is a subdirect product.
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Subgroup of an abelian group consisting of all elements of finite order
In the theory of abelian groups, the torsion subgroup "AT" of an abelian group "A" is the subgroup of "A" consisting of all elements that have finite order (the torsion elements of "A"). An abelian group "A" is called a torsion group (or periodic group) if every element of "A" has finite order and is called torsion-free if every element of "A" except the identity is of infinite order.
The proof that "AT" is closed under the group operation relies on the commutativity of the operation (see examples section).
If "A" is abelian, then the torsion subgroup "T" is a fully characteristic subgroup of "A" and the factor group "A"/"T" is torsion-free. There is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup. There is another covariant functor from the category of abelian groups to the category of torsion-free groups that sends every group to its quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism (which is easily seen to be well-defined).
If "A" is finitely generated and abelian, then it can be written as the direct sum of its torsion subgroup "T" and a torsion-free subgroup (but this is not true for all infinitely generated abelian groups). In any decomposition of "A" as a direct sum of a torsion subgroup "S" and a torsion-free subgroup, "S" must equal "T" (but the torsion-free subgroup is not uniquely determined). This is a key step in the classification of finitely generated abelian groups.
"p"-power torsion subgroups.
For any abelian group $(A, +)$ and any prime number "p" the set "ATp" of elements of "A" that have order a power of "p" is a subgroup called the "p"-power torsion subgroup or, more loosely, the "p"-torsion subgroup:
$A_{T_p}=\{a\in A \;|\; \exists n\in \mathbb{N}\;, p^n a = 0\}.\;$
The torsion subgroup "AT" is isomorphic to the direct sum of its "p"-power torsion subgroups over all prime numbers "p":
$A_T \cong \bigoplus_{p\in P} A_{T_p}.\;$
When "A" is a finite abelian group, "ATp" coincides with the unique Sylow "p"-subgroup of "A".
Each "p"-power torsion subgroup of "A" is a fully characteristic subgroup. More strongly, any homomorphism between abelian groups sends each "p"-power torsion subgroup into the corresponding "p"-power torsion subgroup.
For each prime number "p", this provides a functor from the category of abelian groups to the category of "p"-power torsion groups that sends every group to its "p"-power torsion subgroup, and restricts every homomorphism to the "p"-torsion subgroups. The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of "p"-torsion groups. In a sense, this means that studying "p"-torsion groups in isolation tells us everything about torsion groups in general.
⟨ "x", "y" | "x"² = "y"² = 1 ⟩
the element "xy" is a product of two torsion elements, but has infinite order.
Notes.
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In mathematics, a locally compact group is a topological group "G" for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on "G" so that standard analysis notions such as the Fourier transform and $L^p$ spaces can be generalized.
Many of the results of finite group representation theory are proved by averaging over the group. For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral. In the general locally compact setting, such techniques need not hold. The resulting theory is a central part of harmonic analysis. The representation theory for locally compact abelian groups is described by Pontryagin duality.
Properties.
By homogeneity, local compactness of the underlying space for a topological group need only be checked at the identity. That is, a group "G" is a locally compact space if and only if the identity element has a compact neighborhood. It follows that there is a local base of compact neighborhoods at every point.
A topological group is Hausdorff if and only if the trivial one-element subgroup is closed.
Every closed subgroup of a locally compact group is locally compact. (The closure condition is necessary as the group of rationals demonstrates.) Conversely, every locally compact subgroup of a Hausdorff group is closed. Every quotient of a locally compact group is locally compact. The product of a family of locally compact groups is locally compact if and only if all but a finite number of factors are actually compact.
Topological groups are always completely regular as topological spaces. Locally compact groups have the stronger property of being normal.
Every locally compact group which is first-countable is metrisable as a topological group (i.e. can be given a left-invariant metric compatible with the topology) and complete. If furthermore the space is second-countable, the metric can be chosen to be proper. (See the article on topological groups.)
In a Polish group "G", the σ-algebra of Haar null sets satisfies the countable chain condition if and only if "G" is locally compact.
Locally compact abelian groups.
For any locally compact abelian (LCA) group "A", the group of continuous homomorphisms
Hom("A", "S"1)
from "A" to the circle group is again locally compact. Pontryagin duality asserts that this functor induces an equivalence of categories
LCAop → LCA.
This functor exchanges several properties of topological groups. For example, finite groups correspond to finite groups, compact groups correspond to discrete groups, and metrisable groups correspond to countable unions of compact groups (and vice versa in all statements).
LCA groups form an exact category, with admissible monomorphisms being closed subgroups and admissible epimorphisms being topological quotient maps. It is therefore possible to consider the K-theory spectrum of this category. has shown that it measures the difference between the algebraic K-theory of Z and R, the integers and the reals, respectively, in the sense that there is a homotopy fiber sequence
K(Z) → K(R) → K(LCA).
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A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.
Throughout the article, we use $e$ to denote the identity element of a group.
* !$@ * See also* References
Basic definitions.
Subgroup. A subset $H$ of a group $(G, *)$ which remains a group when the operation $*$ is restricted to $H$ is called a "subgroup" of $G$.
Given a subset $S$ of $G$. We denote by $$ the smallest subgroup of $G$ containing $S$. $$ is called the subgroup of $G$ generated by $S$.
Normal subgroup. $H$ is a "normal subgroup" of $G$ if for all $g$ in $G$ and $h$ in $H$, $g * h * g^{-1}$also belongs to $H$.
Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem.
Group homomorphism. These are functions $f\colon (G,*) \to (H,\times)$ that have the special property that
$ f(a * b) = f(a) \times f(b),$
for any elements $a$ and $b$ of $G$.
Kernel of a group homomorphism. It is the preimage of the identity in the codomain of a group homomorphism. Every normal subgroup is the kernel of a group homomorphism and vice versa.
Group isomorphism. Group homomorphisms that have inverse functions. The inverse of an isomorphism, it turns out, must also be a homomorphism.
Isomorphic groups. Two groups are "isomorphic" if there exists a group isomorphism mapping from one to the other. Isomorphic groups can be thought of as essentially the same, only with different labels on the individual elements.
One of the fundamental problems of group theory is the "classification of groups" up to isomorphism.
Direct product, direct sum, and semidirect product of groups. These are ways of combining groups to construct new groups; please refer to the corresponding links for explanation.
Types of groups.
Finitely generated group. If there exists a finite set $S$ such that $ =G,$ then $G$ is said to be finitely generated. If $S$ can be taken to have just one element, $G$ is a cyclic group of finite order, an infinite cyclic group, or possibly a group $\{e\}$ with just one element.
Simple group. Simple groups are those groups having only $e$ and themselves as normal subgroups. The name is misleading because a simple group can in fact be very complex. An example is the monster group, whose order is about 1054. Every finite group is built up from simple groups via group extensions, so the study of finite simple groups is central to the study of all finite groups. The finite simple groups are known and classified.
The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups.
This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set.
The situation is much more complicated for the non-abelian groups.
Free group. Given any set $A$, one can define a group as the smallest group containing the free semigroup of $A$. The group consists of the finite strings (words) that can be composed by elements from $A$, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance $(abb)*(bca)=abbbca.$
Every group $(G, *)$ is basically a factor group of a free group generated by $G$. Please refer to presentation of a group for more explanation.
One can then ask algorithmic questions about these presentations, such as:
The general case of this is the word problem, and several of these questions are in fact unsolvable by any general algorithm.
General linear group, denoted by GL("n", "F"), is the group of $n$-by-$n$ invertible matrices, where the elements of the matrices are taken from a field $F$ such as the real numbers or the complex numbers.
Group representation (not to be confused with the "presentation" of a group). A "group representation" is a homomorphism from a group to a general linear group. One basically tries to "represent" a given abstract group as a concrete group of invertible matrices which is much easier to study.
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In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer "n" greater than or equal to 4, there are exactly four isomorphism classes of non-abelian groups of order 2"n" which have a cyclic subgroup of index 2. Two are well known, the generalized quaternion group and the dihedral group. One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class. In Bertram Huppert's text "Endliche Gruppen", this group is called a "Quasidiedergruppe". In Daniel Gorenstein's text, "Finite Groups", this group is called the "semidihedral group". Dummit and Foote refer to it as the "quasidihedral group"; we adopt that name in this article. All give the same presentation for this group:
$\langle r,s \mid r^{2^{n-1}} = s^2 = 1,\ srs = r^{2^{n-2}-1}\rangle\,\!$.
The other non-abelian 2-group with cyclic subgroup of index 2 is not given a special name in either text, but referred to as just "G" or M"m"(2). When this group has order 16, Dummit and Foote refer to this group as the "modular group of order 16", as its lattice of subgroups is modular. In this article this group will be called the modular maximal-cyclic group of order $2^n$. Its presentation is:
$\langle r,s \mid r^{2^{n-1}} = s^2 = 1,\ srs = r^{2^{n-2}+1}\rangle\,\!$.
Both these two groups and the dihedral group are semidirect products of a cyclic group <"r"> of order 2"n"−1 with a cyclic group <"s"> of order 2. Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the group of units of the ring $\mathbb{Z}/2^{n-1}\mathbb{Z}$ and there are precisely three such elements, $2^{n-1}-1$, $2^{n-2}-1$, and $2^{n-2}+1$, corresponding to the dihedral group, the quasidihedral, and the modular maximal-cyclic group.
The generalized quaternion group, the dihedral group, and the quasidihedral group of order 2"n" all have nilpotency class "n" − 1, and are the only isomorphism classes of groups of order 2"n" with nilpotency class "n" − 1. The groups of order "p""n" and nilpotency class "n" − 1 were the beginning of the classification of all "p"-groups via coclass. The modular maximal-cyclic group of order 2"n" always has nilpotency class 2. This makes the modular maximal-cyclic group less interesting, since most groups of order "p""n" for large "n" have nilpotency class 2 and have proven difficult to understand directly.
The generalized quaternion, the dihedral, and the quasidihedral group are the only 2-groups whose derived subgroup has index 4. The Alperin–Brauer–Gorenstein theorem classifies the simple groups, and to a degree the finite groups, with quasidihedral Sylow 2-subgroups.
Examples.
The Sylow 2-subgroups of the following groups are quasidihedral:
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In mathematical group theory, Frobenius's theorem states that if "n" divides the order of a finite group "G", then the number of solutions of "x""n" = 1 is a multiple of "n". It was introduced by Frobenius (1903).
Statement.
A more general version of Frobenius's theorem states that if "C" is a conjugacy class with "h" elements of a finite group "G" with "g" elements and "n" is a positive integer, then the number of elements "k" such that "k""n" is in "C" is a multiple of the greatest common divisor ("hn","g") .
Applications.
One application of Frobenius's theorem is to show that the coefficients of the Artin–Hasse exponential are "p" integral, by interpreting them in terms of the number of elements of order a power of "p" in the symmetric group "S""n".
Frobenius conjecture.
Frobenius conjectured that if in addition the number of solutions to "x""n"=1 is exactly "n" where "n" divides the order of "G" then these solutions form a normal subgroup. This has been proved as a consequence of the classification of finite simple groups. The symmetric group "S"3 has exactly 4 solutions to "x"4=1 but these do not form a normal subgroup; this is not a counterexample to the conjecture as 4 does not divide the order of "S"3.
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Subset of a group that forms a group itself
In group theory, a branch of mathematics, given a group "G" under a binary operation ∗, a subset "H" of "G" is called a subgroup of "G" if "H" also forms a group under the operation ∗. More precisely, "H" is a subgroup of "G" if the restriction of ∗ to "H" × "H" is a group operation on "H". This is often denoted "H" ≤ "G", read as ""H" is a subgroup of "G"".
The trivial subgroup of any group is the subgroup {"e"} consisting of just the identity element.
A proper subgroup of a group "G" is a subgroup "H" which is a proper subset of "G" (that is, "H" ≠ "G"). This is often represented notationally by "H" < "G", read as ""H" is a proper subgroup of "G"". Some authors also exclude the trivial group from being proper (that is, "H" ≠ {"e"}).
If "H" is a subgroup of "G", then "G" is sometimes called an overgroup of "H".
The same definitions apply more generally when "G" is an arbitrary semigroup, but this article will only deal with subgroups of groups.
Subgroup tests.
Suppose that "G" is a group, and "H" is a subset of "G". For now, assume that the group operation of "G" is written multiplicatively, denoted by juxtaposition.
If the group operation is instead denoted by addition, then "closed under products" should be replaced by "closed under addition", which is the condition that for every "a" and "b" in "H", the sum "a"+"b" is in "H", and "closed under inverses" should be edited to say that for every "a" in "H", the inverse −"a" is in "H".
Cosets and Lagrange's theorem.
Given a subgroup "H" and some "a" in G, we define the left coset "aH" = {"ah" : "h" in "H"}. Because "a" is invertible, the map φ : "H" → "aH" given by φ("h") = "ah" is a bijection. Furthermore, every element of "G" is contained in precisely one left coset of "H"; the left cosets are the equivalence classes corresponding to the equivalence relation "a"1 ~ "a"2 if and only if "a"1−1"a"2 is in "H". The number of left cosets of "H" is called the index of "H" in "G" and is denoted by ["G" : "H"].
Lagrange's theorem states that for a finite group "G" and a subgroup "H",
$ [ G : H ] = { |G| \over |H| }$
where |"G"| and |"H"| denote the orders of "G" and "H", respectively. In particular, the order of every subgroup of "G" (and the order of every element of "G") must be a divisor of |"G"|.
Right cosets are defined analogously: "Ha" = {"ha" : "h" in "H"}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to ["G" : "H"].
If "aH" = "Ha" for every "a" in "G", then "H" is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if "p" is the lowest prime dividing the order of a finite group "G," then any subgroup of index "p" (if such exists) is normal.
Example: Subgroups of Z8.
Let "G" be the cyclic group Z8 whose elements are
$G = \left\{0, 4, 2, 6, 1, 5, 3, 7\right\}$
and whose group operation is addition modulo 8. Its Cayley table is
This group has two nontrivial subgroups: and , where "J" is also a subgroup of "H". The Cayley table for "H" is the top-left quadrant of the Cayley table for "G"; The Cayley table for "J" is the top-left quadrant of the Cayley table for "H". The group "G" is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
Example: Subgroups of S4.
S4 is the symmetric group whose elements correspond to the permutations of 4 elements.<br>
Below are all its subgroups, ordered by cardinality.<br>
Each group is represented by its Cayley table.
24 elements.
Like each group, S4 is a subgroup of itself.
12 elements.
The alternating group contains only the even permutations.<br>
It is one of the two nontrivial proper normal subgroups of S4.
2 elements.
Each permutation p of order 2 generates a subgroup {1, "p"}.
These are the permutations that have only 2-cycles:<br>
1 element.
The trivial subgroup is the unique subgroup of order 1.
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abstract_algebra
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In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let "G" be a group, and let "H" and "K" be subgroups. Let "H" act on "G" by left multiplication and let "K" act on "G" by right multiplication. For each "x" in "G", the ("H", "K")-double coset of "x" is the set
$HxK = \{ hxk \colon h \in H, k \in K \}.$
When "H" = "K", this is called the "H"-double coset of "x". Equivalently, "HxK" is the equivalence class of "x" under the equivalence relation
"x" ~ "y" if and only if there exist "h" in "H" and "k" in "K" such that "hxk" = "y".
The set of all $(H,K)$-double cosets is denoted by $H \,\backslash G / K.$
Properties.
Suppose that "G" is a group with subgroups "H" and "K" acting by left and right multiplication, respectively. The ("H", "K")-double cosets of "G" may be equivalently described as orbits for the product group "H" × "K" acting on "G" by ("h", "k") ⋅ "x" = "hxk"−1. Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because "G" is a group and "H" and "K" are subgroups acting by multiplication, double cosets are more structured than orbits of arbitrary group actions, and they have additional properties that are false for more general actions.
HxK &= \bigcup_{k \in K} Hxk = \coprod_{Hxk \,\in\, H \backslash HxK} Hxk,
HxK &= \bigcup_{h \in H} hxK = \coprod_{hxK \,\in\, HxK / K} hxK.
\end{align}$
\left[G : H\right] &= \sum_{HxK \,\in\, H \backslash G / K} [K : K \cap x^{-1}Hx], \\
\left[G : K\right] &= \sum_{HxK \,\in\, H \backslash G / K} [H : H \cap xKx^{-1}].
\end{align}$
\left[G : H\right] &= \sum_{HxK \,\in\, H \backslash G / K} \frac, \\
\left[G : K\right] &= \sum_{HxK \,\in\, H \backslash G / K} \frac.
\end{align}$
There is an equivalent description of double cosets in terms of single cosets. Let "H" and "K" both act by right multiplication on "G". Then "G" acts by left multiplication on the product of coset spaces "G" / "H" × "G" / "K". The orbits of this action are in one-to-one correspondence with "H" \ "G" / "K". This correspondence identifies ("xH", "yK") with the double coset "Hx"−1"yK". Briefly, this is because every "G"-orbit admits representatives of the form ("H", "xK"), and the representative "x" is determined only up to left multiplication by an element of "H". Similarly, "G" acts by right multiplication on "H" \ "G" × "K" \ "G", and the orbits of this action are in one-to-one correspondence with the double cosets "H" \ "G" / "K". Conceptually, this identifies the double coset space "H" \ "G" / "K" with the space of relative configurations of an "H"-coset and a "K"-coset. Additionally, this construction generalizes to the case of any number of subgroups. Given subgroups "H"1, ..., "H""n", the space of ("H"1, ..., "H""n")-multicosets is the set of "G"-orbits of "G" / "H"1 × ... × "G" / "H""n".
The analog of Lagrange's theorem for double cosets is false. This means that the size of a double coset need not divide the order of "G". For example, let "G" = "S"3 be the symmetric group on three letters, and let "H" and "K" be the cyclic subgroups generated by the transpositions (1 2) and (1 3), respectively. If "e" denotes the identity permutation, then
$HeK = HK = \{ e, (1 2), (1 3), (1 3 2) \}.$
This has four elements, and four does not divide six, the order of "S"3. It is also false that different double cosets have the same size. Continuing the same example,
$H(2 3)K = \{ (2 3), (1 2 3) \},$
which has two elements, not four.
However, suppose that "H" is normal. As noted earlier, in this case the double coset space equals the left coset space "G" / "HK". Similarly, if "K" is normal, then "H" \ "G" / "K" is the right coset space "HK" \ "G". Standard results about left and right coset spaces then imply the following facts.
Products in the free abelian group on the set of double cosets.
Suppose that "G" is a group and that "H", "K", and "L" are subgroups. Under certain finiteness conditions, there is a product on the free abelian group generated by the ("H", "K")- and ("K", "L")-double cosets with values in the free abelian group generated by the ("H", "L")-double cosets. This means there is a bilinear function
$\mathbf{Z}[H \backslash G / K] \times \mathbf{Z}[K \backslash G / L] \to \mathbf{Z}[H \backslash G / L].$
Assume for simplicity that "G" is finite. To define the product, reinterpret these free abelian groups in terms of the group algebra of "G" as follows. Every element of Z["H" \ "G" / "K"] has the form
$\sum_{HxK \in H \backslash G / K} f_{HxK} \cdot [HxK],$
where { "f""HxK" } is a set of integers indexed by the elements of "H" \ "G" / "K". This element may be interpreted as a Z-valued function on "H" \ "G" / "K", specifically, "HxK" ↦ "f""HxK". This function may be pulled back along the projection "G" → "H" \ "G" / "K" which sends "x" to the double coset "HxK". This results in a function "x" ↦ "f""HxK". By the way in which this function was constructed, it is left invariant under "H" and right invariant under "K". The corresponding element of the group algebra Z["G"] is
$\sum_{x \in G} f_{HxK} \cdot [x],$
and this element is invariant under left multiplication by "H" and right multiplication by "K". Conceptually, this element is obtained by replacing "HxK" by the elements it contains, and the finiteness of "G" ensures that the sum is still finite. Conversely, every element of Z["G"] which is left invariant under "H" and right invariant under "K" is the pullback of a function on Z["H" \ "G" / "K"]. Parallel statements are true for Z["K" \ "G" / "L"] and Z["H" \ "G" / "L"].
When elements of Z["H" \ "G" / "K"], Z["K" \ "G" / "L"], and Z["H" \ "G" / "L"] are interpreted as invariant elements of Z["G"], then the product whose existence was asserted above is precisely the multiplication in Z["G"]. Indeed, it is trivial to check that the product of a left-"H"-invariant element and a right-"L"-invariant element continues to be left-"H"-invariant and right-"L"-invariant. The bilinearity of the product follows immediately from the bilinearity of multiplication in Z["G"]. It also follows that if "M" is a fourth subgroup of "G", then the product of ("H", "K")-, ("K", "L")-, and ("L", "M")-double cosets is associative. Because the product in Z["G"] corresponds to convolution of functions on "G", this product is sometimes called the convolution product.
An important special case is when "H" = "K" = "L". In this case, the product is a bilinear function
$\mathbf{Z}[H \backslash G / H] \times \mathbf{Z}[H \backslash G / H] \to \mathbf{Z}[H \backslash G / H].$
This product turns Z["H" \ "G" / "H"] into an associative ring whose identity element is the class of the trivial double coset ["H"]. In general, this ring is non-commutative. For example, if "H" = {1}, then the ring is the group algebra Z["G"], and a group algebra is a commutative ring if and only if the underlying group is abelian.
If "H" is normal, so that the "H"-double cosets are the same as the elements of the quotient group "G" / "H", then the product on Z["H" \ "G" / "H"] is the product in the group algebra Z["G" / "H"]. In particular, it is the usual convolution of functions on "G" / "H". In this case, the ring is commutative if and only if "G" / "H" is abelian, or equivalently, if and only if "H" contains the commutator subgroup of "G".
If "H" is not normal, then Z["H" \ "G" / "H"] may be commutative even if "G" is non-abelian. A classical example is the product of two Hecke operators. This is the product in the Hecke algebra, which is commutative even though the group "G" is the modular group, which is non-abelian, and the subgroup is an arithmetic subgroup and in particular does not contain the commutator subgroup. Commutativity of the convolution product is closely tied to Gelfand pairs.
When the group "G" is a topological group, it is possible to weaken the assumption that the number of left and right cosets in each double coset is finite. The group algebra Z["G"] is replaced by an algebra of functions such as "L"2("G") or "C"∞("G"), and the sums are replaced by integrals. The product still corresponds to convolution. For instance, this happens for the Hecke algebra of a locally compact group.
Applications.
When a group $G $ has a transitive group action on a set $S$, computing certain double coset decompositions of $G $ reveals extra information about structure of the action of $G $ on $S $. Specifically, if $H $ is the stabilizer subgroup of some element $s\in S $, then $G $ decomposes as exactly two double cosets of $(H,H) $ if and only if $G $ acts transitively on the set of distinct pairs of $S$. See 2-transitive groups for more information about this action.
Double cosets are important in connection with representation theory, when a representation of "H" is used to construct an induced representation of "G", which is then restricted to "K". The corresponding double coset structure carries information about how the resulting representation decomposes. In the case of finite groups, this is Mackey's decomposition theorem.
They are also important in functional analysis, where in some important cases functions left-invariant and right-invariant by a subgroup "K" can form a commutative ring under convolution: see Gelfand pair.
In geometry, a Clifford–Klein form is a double coset space Γ\"G"/"H", where "G" is a reductive Lie group, "H" is a closed subgroup, and Γ is a discrete subgroup (of "G") that acts properly discontinuously on the homogeneous space "G"/"H".
In number theory, the Hecke algebra corresponding to a congruence subgroup "Γ" of the modular group is spanned by elements of the double coset space $\Gamma \backslash \mathrm{GL}_2^+(\mathbb{Q}) / \Gamma$; the algebra structure is that acquired from the multiplication of double cosets described above. Of particular importance are the Hecke operators $T_m$ corresponding to the double cosets $\Gamma_0(N) g \Gamma_0(N)$ or $\Gamma_1(N) g \Gamma_1(N)$, where $g= \left( \begin{smallmatrix} 1 & 0 \\ 0 & m \end{smallmatrix} \right)$ (these have different properties depending on whether "m" and "N" are coprime or not), and the diamond operators $ \langle d \rangle$ given by the double cosets $ \Gamma_1(N) \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \Gamma_1(N)$ where $ d \in (\mathbb{Z}/N\mathbb{Z})^\times$ and we require $ \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)\in \Gamma_0(N)$ (the choice of "a", "b", "c" does not affect the answer).
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abstract_algebra
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Theorem that every subgroup of a free group is itself free
In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier.
Statement of the theorem.
A free group may be defined from a group presentation consisting of a set of generators with no relations. That is, every element is a product of some sequence of generators and their inverses, but these elements do not obey any equations except those trivially following from "gg"−1 = 1. The elements of a free group may be described as all possible reduced words, those strings of generators and their inverses in which no generator is adjacent to its own inverse. Two reduced words may be multiplied by concatenating them and then removing any generator-inverse pairs that result from the concatenation.
The Nielsen–Schreier theorem states that if "H" is a subgroup of a free group "G", then "H" is itself isomorphic to a free group. That is, there exists a set "S" of elements which generate "H", with no nontrivial relations among the elements of "S".
The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the subgroup has finite index: if "G" is a free group of rank "n" (free on "n" generators), and "H" is a subgroup of finite index ["G" : "H"] = "e", then "H" is free of rank $ 1 + e(n{-}1) $.
Example.
Let "G" be the free group with two generators $a,b$, and let "H" be the subgroup consisting of all reduced words of even length (products of an even number of letters $ a,b,a^{-1},b^{-1} $). Then "H" is generated by its six elements $p=aa,\ q=ab,\ r=ba,\ s=bb,\ t=ab^{-1},\ u=a^{-1}b.$ A factorization of any reduced word in "H" into these generators and their inverses may be constructed simply by taking consecutive pairs of letters in the reduced word. However, this is not a free presentation of "H" because the last three generators can be written in terms of the first three as $s=rp^{-1}q,\ t=pr^{-1},\ u=p^{-1}q$. Rather, "H" is generated as a free group by the three elements $p=aa,\ q=ab,\ r=ba,$ which have no relations among them; or instead by several other triples of the six generators. Further, "G" is free on "n" = 2 generators, "H" has index "e" = ["G" : "H"] = 2 in "G", and "H" is free on 1 + "e"("n"–1) = 3 generators. The Nielsen–Schreier theorem states that like "H", every subgroup of a free group can be generated as a free group, and if the index of "H" is finite, its rank is given by the index formula.
Proof.
A short proof of the Nielsen–Schreier theorem uses the algebraic topology of fundamental groups and covering spaces. A free group "G" on a set of generators is the fundamental group of a bouquet of circles, a topological graph "X" with a single vertex and with a loop-edge for each generator. Any subgroup "H" of the fundamental group is itself the fundamental group of a connected covering space "Y" → "X." The space "Y" is a (possibly infinite) topological graph, the Schreier coset graph having one vertex for each coset in "G/H". In any connected topological graph, it is possible to shrink the edges of a spanning tree of the graph, producing a bouquet of circles that has the same fundamental group "H". Since "H" is the fundamental group of a bouquet of circles, it is itself free.
Simplicial homology allows the computation of the rank of "H", which is equal to "h"1("Y"), the first Betti number of the covering space, the number of independent cycles. For "G" free of rank "n", the graph "X" has "n" edges and 1 vertex; assuming "H" has finite index ["G" : "H"] = "e", the covering graph "Y" has "en" edges and "e" vertices. The first Betti number of a graph is equal to the number of edges, minus the number of vertices, plus the number of connected components; hence the rank of "H" is:
$h_1(Y) \,=\, en-e+1 \,=\, 1+e(n{-}1).$
This proof is due to Reinhold Baer and Friedrich Levi (1936); the original proof by Schreier forms the Schreier graph in a different way as a quotient of the Cayley graph of G modulo the action of H.
According to Schreier's subgroup lemma, a set of generators for a free presentation of H may be constructed from cycles in the covering graph formed by concatenating a spanning tree path from a base point (the coset of the identity) to one of the cosets, a single non-tree edge, and an inverse spanning tree path from the other endpoint of the edge back to the base point.
Axiomatic foundations.
Although several different proofs of the Nielsen–Schreier theorem are known, they all depend on the axiom of choice. In the proof based on fundamental groups of bouquets, for instance, the axiom of choice appears in the guise of the statement that every connected graph has a spanning tree. The use of this axiom is necessary, as there exist models of Zermelo–Fraenkel set theory in which the axiom of choice and the Nielsen–Schreier theorem are both false. The Nielsen–Schreier theorem in turn implies a weaker version of the axiom of choice, for finite sets.
History.
The Nielsen–Schreier theorem is a non-abelian analogue of an older result of Richard Dedekind, that every subgroup of a free abelian group is free abelian.
Jakob Nielsen (1921) originally proved a restricted form of the theorem, stating that any finitely-generated subgroup of a free group is free. His proof involves performing a sequence of Nielsen transformations on the subgroup's generating set that reduce their length (as reduced words in the free group from which they are drawn). Otto Schreier proved the Nielsen–Schreier theorem in its full generality in his 1926 habilitation thesis, "Die Untergruppen der freien Gruppe", also published in 1927 in "Abh. math. Sem. Hamburg. Univ."
The topological proof based on fundamental groups of bouquets of circles is due to Reinhold Baer and Friedrich Levi (1936). Another topological proof, based on the Bass–Serre theory of group actions on trees, was published by Jean-Pierre Serre (1970).
Notes.
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2228143
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abstract_algebra
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Disjoint, equal-size subsets of a group's underlying set
In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets. There are "left cosets" and "right cosets". Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by ["G" : "H"].
Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G. Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.
Definition.
Let H be a subgroup of the group G whose operation is written multiplicatively (juxtaposition denotes the group operation). Given an element g of G, the left cosets of H in G are the sets obtained by multiplying each element of H by a fixed element g of G (where g is the left factor). In symbols these are,
"gH" = {"gh" : "h" an element of "H"} for g in G.
The right cosets are defined similarly, except that the element g is now a right factor, that is,
"Hg" = {"hg" : "h" an element of "H"} for g in G.
As g varies through the group, it would appear that many cosets (right or left) would be generated. Nevertheless, it turns out that any two left cosets (respectively right cosets) are either disjoint or are identical as sets.
If the group operation is written additively, as is often the case when the group is abelian, the notation used changes to "g" + "H" or "H" + "g", respectively.
First example.
Let G be the dihedral group of order six. Its elements may be represented by {"I", "a", "a"2, "b", "ab", "a"2"b"}. In this group, "a"3 = "b"2 = "I" and "ba" = "a"2"b". This is enough information to fill in the entire Cayley table:
Let T be the subgroup {"I", "b"}. The (distinct) left cosets of T are:
Since all the elements of G have now appeared in one of these cosets, generating any more can not give new cosets, since a new coset would have to have an element in common with one of these and therefore be identical to one of these cosets. For instance, "abT" = {"ab", "a"} = "aT".
The right cosets of T are:
In this example, except for T, no left coset is also a right coset.
Let H be the subgroup {"I", "a", "a"2}. The left cosets of H are "IH" = "H" and "bH" = {"b", "ba", "ba"2}. The right cosets of H are "HI" = "H" and "Hb" = {"b", "ab", "a"2"b"} = {"b", "ba"2, "ba"}. In this case, every left coset of H is also a right coset of H.
Let "H" be a subgroup of a group "G" and suppose that "g"1, "g"2 ∈ "G". The following statements are equivalent:
Properties.
The disjointness of non-identical cosets is a result of the fact that if x belongs to "gH" then "gH" = "xH". For if "x" ∈ "gH" then there must exist an "a" ∈ "H" such that "ga" = "x". Thus "xH" = ("ga")"H" = "g"("aH"). Moreover, since "H" is a group, left multiplication by a is a bijection, and "aH" = "H".
Thus every element of "G" belongs to exactly one left coset of the subgroup "H", and "H" is itself a left coset (and the one that contains the identity).
Two elements being in the same left coset also provide a natural equivalence relation. Define two elements of G, x and y, to be equivalent with respect to the subgroup H if "xH" = "yH" (or equivalently if "x"−1"y" belongs to H). The equivalence classes of this relation are the left cosets of H. As with any set of equivalence classes, they form a partition of the underlying set. A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here.
Similar statements apply to right cosets.
If "G" is an abelian group, then "g" + "H" = "H" + "g" for every subgroup "H" of "G" and every element g of "G". For general groups, given an element g and a subgroup "H" of a group "G", the right coset of "H" with respect to g is also the left coset of the conjugate subgroup "g"−1"Hg" with respect to g, that is, "Hg" = "g"("g"−1"Hg").
Normal subgroups.
A subgroup "N" of a group "G" is a normal subgroup of "G" if and only if for all elements g of "G" the corresponding left and right cosets are equal, that is, "gN" = "Ng". This is the case for the subgroup H in the first example above. Furthermore, the cosets of "N" in "G" form a group called the quotient group or factor group "G"/"N".
If "H" is not normal in "G", then its left cosets are different from its right cosets. That is, there is an a in "G" such that no element b satisfies "aH" = "Hb". This means that the partition of "G" into the left cosets of "H" is a different partition than the partition of "G" into right cosets of "H". This is illustrated by the subgroup T in the first example above. ("Some" cosets may coincide. For example, if a is in the center of "G", then "aH" = "Ha".)
On the other hand, if the subgroup "N" is normal the set of all cosets forms a group called the quotient group "G" / "N" with the operation ∗ defined by ("aN") ∗ ("bN") = "abN". Since every right coset is a left coset, there is no need to distinguish "left cosets" from "right cosets".
Index of a subgroup.
Every left or right coset of "H" has the same number of elements (or cardinality in the case of an infinite "H") as "H" itself. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of "H" in "G", written as ["G" : "H"]. Lagrange's theorem allows us to compute the index in the case where "G" and "H" are finite:
<math display="block">|G| = [G : H]|H|.$
This equation also holds in the case where the groups are infinite, although the meaning may be less clear.
More examples.
Integers.
Let "G" be the additive group of the integers, Z = ({..., −2, −1, 0, 1, 2, ...}, +) and "H" the subgroup (3Z, +) = ({..., −6, −3, 0, 3, 6, ...}, +). Then the cosets of "H" in "G" are the three sets 3Z, 3Z + 1, and 3Z + 2, where 3Z + "a" = {..., −6 + "a", −3 + "a", "a", 3 + "a", 6 + "a", ...}. These three sets partition the set Z, so there are no other right cosets of H. Due to the commutivity of addition "H" + 1 = 1 + "H" and "H" + 2 = 2 + "H". That is, every left coset of H is also a right coset, so H is a normal subgroup. (The same argument shows that every subgroup of an Abelian group is normal.)
This example may be generalized. Again let "G" be the additive group of the integers, Z = ({..., −2, −1, 0, 1, 2, ...}, +), and now let "H" the subgroup ("mZ, +) = ({..., −2"m", −"m", 0, "m", 2"m", ...}, +), where m is a positive integer. Then the cosets of "H" in "G" are the m sets "mZ, "mZ + 1, ..., "mZ + ("m" − 1), where "mZ + "a" = {..., −2"m"+"a", −"m"+"a", "a", "m"+"a", 2"m"+"a", ...}. There are no more than m cosets, because "mZ + "m" = "m"(Z + 1) = "mZ. The coset ("mZ + "a", +) is the congruence class of a modulo m. The subgroup mZ is normal in Z, and so, can be used to form the quotient group Z/"m"Z the group of integers mod "m".
Vectors.
Another example of a coset comes from the theory of vector spaces. The elements (vectors) of a vector space form an abelian group under vector addition. The subspaces of the vector space are subgroups of this group. For a vector space "V", a subspace "W", and a fixed vector a in "V", the sets
<math display="block">\{\mathbf{x} \in V \mid \mathbf{x} = \mathbf{a} + \mathbf{w}, \mathbf{w} \in W\}$
are called affine subspaces, and are cosets (both left and right, since the group is abelian). In terms of 3-dimensional geometric vectors, these affine subspaces are all the "lines" or "planes" parallel to the subspace, which is a line or plane going through the origin. For example, consider the plane R2. If m is a line through the origin O, then m is a subgroup of the abelian group R2. If P is in R2, then the coset "P" + "m" is a line "m"′ parallel to m and passing through P.
Matrices.
Let G be the multiplicative group of matrices,
<math display="block">G = \left \{\begin{bmatrix} a & 0 \\ b & 1 \end{bmatrix} \colon a, b \in \R, a \neq 0 \right\},$
and the subgroup H of G,
<math display="block">H= \left \{\begin{bmatrix} 1 & 0 \\ c & 1 \end{bmatrix} \colon c \in \mathbb{R} \right\}.$
For a fixed element of G consider the left coset
\begin{bmatrix} a & 0 \\ b & 1 \end{bmatrix} H = &~ \left \{\begin{bmatrix} a & 0 \\ b & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ c & 1 \end{bmatrix} \colon c \in \R \right\} \\
=&~ \left \{\begin{bmatrix} a & 0 \\ b + c & 1 \end{bmatrix} \colon c \in \mathbb{R}\right\} \\
=&~ \left \{\begin{bmatrix} a & 0 \\ d & 1 \end{bmatrix} \colon d \in \mathbb{R}\right\}.
\end{align}$
That is, the left cosets consist of all the matrices in G having the same upper-left entry. This subgroup H is normal in G, but the subgroup
<math display="block">T= \left \{\begin{bmatrix} a & 0 \\ 0 & 1 \end{bmatrix} \colon a \in \mathbb{R} - \{0\} \right\}$
is not normal in G.
As orbits of a group action.
A subgroup H of a group G can be used to define an action of H on G in two natural ways. A "right action", "G" × "H" → "G" given by ("g", "h") → "gh" or a "left action", "H" × "G" → "G" given by ("h", "g") → "hg". The orbit of g under the right action is the left coset gH, while the orbit under the left action is the right coset Hg.
History.
The concept of a coset dates back to Galois's work of 1830–31. He introduced a notation but did not provide a name for the concept. The term "co-set" appears for the first time in 1910 in a paper by G. A. Miller in the "Quarterly Journal of Mathematics" (vol. 41, p. 382). Various other terms have been used including the German "Nebengruppen" (Weber) and "conjugate group" (Burnside).
Galois was concerned with deciding when a given polynomial equation was solvable by radicals. A tool that he developed was in noting that a subgroup H of a group of permutations G induced two decompositions of G (what we now call left and right cosets). If these decompositions coincided, that is, if the left cosets are the same as the right cosets, then there was a way to reduce the problem to one of working over H instead of G. Camille Jordan in his commentaries on Galois's work in 1865 and 1869 elaborated on these ideas and defined normal subgroups as we have above, although he did not use this term.
Calling the coset gH the "left coset" of g with respect to H, while most common today, has not been universally true in the past. For instance, would call gH a "right coset", emphasizing the subgroup being on the right.
An application from coding theory.
A binary linear code is an n-dimensional subspace C of an m-dimensional vector space V over the binary field GF(2). As V is an additive abelian group, C is a subgroup of this group. Codes can be used to correct errors that can occur in transmission. When a "codeword" (element of C) is transmitted some of its bits may be altered in the process and the task of the receiver is to determine the most likely codeword that the corrupted "received word" could have started out as. This procedure is called "decoding" and if only a few errors are made in transmission it can be done effectively with only a very few mistakes. One method used for decoding uses an arrangement of the elements of V (a received word could be any element of V) into a standard array. A standard array is a coset decomposition of V put into tabular form in a certain way. Namely, the top row of the array consists of the elements of C, written in any order, except that the zero vector should be written first. Then, an element of V with a minimal number of ones that does not already appear in the top row is selected and the coset of C containing this element is written as the second row (namely, the row is formed by taking the sum of this element with each element of C directly above it). This element is called a coset leader and there may be some choice in selecting it. Now the process is repeated, a new vector with a minimal number of ones that does not already appear is selected as a new coset leader and the coset of C containing it is the next row. The process ends when all the vectors of V have been sorted into the cosets.
An example of a standard array for the 2-dimensional code "C" = {00000, 01101, 10110, 11011} in the 5-dimensional space V (with 32 vectors) is as follows:
The decoding procedure is to find the received word in the table and then add to it the coset leader of the row it is in. Since in binary arithmetic adding is the same operation as subtracting, this always results in an element of C. In the event that the transmission errors occurred in precisely the non-zero positions of the coset leader the result will be the right codeword. In this example, if a single error occurs, the method will always correct it, since all possible coset leaders with a single one appear in the array.
Syndrome decoding can be used to improve the efficiency of this method. It is a method of computing the correct coset (row) that a received word will be in. For an n-dimensional code C in an m-dimensional binary vector space, a parity check matrix is an ("m" − "n") × "m" matrix H having the property that xH"T = 0 if and only if x is in C. The vector xH"T is called the "syndrome" of x, and by linearity, every vector in the same coset will have the same syndrome. To decode, the search is now reduced to finding the coset leader that has the same syndrome as the received word.
Double cosets.
Given two subgroups, "H" and "K" (which need not be distinct) of a group "G", the double cosets of "H" and "K" in "G" are the sets of the form "HgK" = {"hgk" : "h" an element of "H", "k" an element of "K"}. These are the left cosets of "K" and right cosets of "H" when "H" = 1 and "K" = 1 respectively.
Two double cosets "HxK" and "HyK" are either disjoint or identical. The set of all double cosets for fixed H and K form a partition of G.
A double coset "HxK" contains the complete right cosets of H (in G) of the form "Hxk", with k an element of K and the complete left cosets of K (in G) of the form "hxK", with h in H.
Notation.
Let "G" be a group with subgroups "H" and "K". Several authors working with these sets have developed a specialized notation for their work, where
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Commutative group (mathematics)
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.
Definition.
An abelian group is a set $A$, together with an operation $\cdot$ that combines any two elements $a$ and $b$ of $A$ to form another element of $A,$ denoted $a \cdot b$. The symbol $\cdot$ is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, $(A, \cdot)$, must satisfy four requirements known as the "abelian group axioms" (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is "defined" for any ordered pair of elements of A, that the result is "well-defined", and that the result "belongs to" A):
A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".
Facts.
Notation.
There are two main notational conventions for abelian groups – additive and multiplicative.
Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules and rings. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being near-rings and partially ordered groups, where an operation is written additively even when non-abelian.
Multiplication table.
To verify that a finite group is abelian, a table (matrix) – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is $G = \{g_1 = e, g_2, \dots, g_n \}$ under the operation $\cdot$, the $(i, j)$-th entry of this table contains the product $g_i \cdot g_j$.
The group is abelian if and only if this table is symmetric about the main diagonal. This is true since the group is abelian iff $g_i \cdot g_j = g_j \cdot g_i$ for all $i, j = 1, ..., n$, which is iff the $(i, j)$ entry of the table equals the $(j, i)$ entry for all $i, j = 1, ..., n$, i.e. the table is symmetric about the main diagonal.
Examples.
In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of $2 \times 2$ rotation matrices.
Historical remarks.
Camille Jordan named abelian groups after Norwegian mathematician Niels Henrik Abel, as Abel had found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals.
Properties.
If $n$ is a natural number and $x$ is an element of an abelian group $G$ written additively, then $nx$ can be defined as $x + x + \cdots + x$ ($n$ summands) and $(-n)x = -(nx)$. In this way, $G$ becomes a module over the ring $\mathbb{Z}$ of integers. In fact, the modules over $\mathbb{Z}$ can be identified with the abelian groups.
Theorems about abelian groups (i.e. modules over the principal ideal domain $\mathbb{Z}$) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form $\mathbb{Z}/p^k\mathbb{Z}$ for $p$ prime, and the latter is a direct sum of finitely many copies of $\mathbb{Z}$.
If $f, g: G \to H$ are two group homomorphisms between abelian groups, then their sum $f + g$, defined by $(f + g)(x) = f(x) + g(x)$, is again a homomorphism. (This is not true if $H$ is a non-abelian group.) The set $\text{Hom}(G,H)$ of all group homomorphisms from $G$ to $H$ is therefore an abelian group in its own right.
Somewhat akin to the dimension of vector spaces, every abelian group has a "rank". It is defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group. Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero is a torsion group. The integers and the rational numbers have rank one, as well as every nonzero additive subgroup of the rationals. On the other hand, the multiplicative group of the nonzero rationals has an infinite rank, as it is a free abelian group with the set of the prime numbers as a basis (this results from the fundamental theorem of arithmetic).
The center $Z(G)$ of a group $G$ is the set of elements that commute with every element of $G$. A group $G$ is abelian if and only if it is equal to its center $Z(G)$. The center of a group $G$ is always a characteristic abelian subgroup of $G$. If the quotient group $G/Z(G)$ of a group by its center is cyclic then $G$ is abelian.
Finite abelian groups.
Cyclic groups of integers modulo $n$, $\mathbb{Z}/n\mathbb{Z}$, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.
Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is also abelian. In fact, for every prime number $p$ there are (up to isomorphism) exactly two groups of order $p^2$, namely $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p\times\mathbb{Z}_p$.
Classification.
The fundamental theorem of finite abelian groups states that every finite abelian group $G$ can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. This is generalized by the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when "G" has zero rank; this in turn admits numerous further generalizations.
The classification was proven by Leopold Kronecker in 1870, though it was not stated in modern group-theoretic terms until later, and was preceded by a similar classification of quadratic forms by Carl Friedrich Gauss in 1801; see history for details.
The cyclic group $\mathbb{Z}_{mn}$ of order $mn$ is isomorphic to the direct sum of $\mathbb{Z}_m$ and $\mathbb{Z}_n$ if and only if $m$ and $n$ are coprime. It follows that any finite abelian group $G$ is isomorphic to a direct sum of the form
$\bigoplus_{i=1}^{u}\ \mathbb{Z}_{k_i}$
in either of the following canonical ways:
For example, $\mathbb{Z}_{15}$ can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: $\mathbb{Z}_{15} \cong \{0,5,10\} \oplus \{0,3,6,9,12\}$. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic.
For another example, every abelian group of order 8 is isomorphic to either $\mathbb{Z}_8$ (the integers 0 to 7 under addition modulo 8), $\mathbb{Z}_4\oplus \mathbb{Z}_2$ (the odd integers 1 to 15 under multiplication modulo 16), or $\mathbb{Z}_2\oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$.
See also list of small groups for finite abelian groups of order 30 or less.
Automorphisms.
One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group $G$. To do this, one uses the fact that if $G$ splits as a direct sum $H\oplus K$ of subgroups of coprime order, then
$\operatorname{Aut}(H\oplus K) \cong \operatorname{Aut}(H)\oplus \operatorname{Aut}(K).$
Given this, the fundamental theorem shows that to compute the automorphism group of $G$ it suffices to compute the automorphism groups of the Sylow $p$-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of $p$). Fix a prime $p$ and suppose the exponents $e_i$ of the cyclic factors of the Sylow $p$-subgroup are arranged in increasing order:
$e_1\leq e_2 \leq\cdots\leq e_n$
for some $n > 0$. One needs to find the automorphisms of
$\mathbf{Z}_{p^{e_1}} \oplus \cdots \oplus \mathbf{Z}_{p^{e_n}}.$
One special case is when $n = 1$, so that there is only one cyclic prime-power factor in the Sylow $p$-subgroup $P$. In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when $n$ is arbitrary but $e_i = 1$ for $ 1 \le i \le n$. Here, one is considering $P$ to be of the form
$\mathbf{Z}_p \oplus \cdots \oplus \mathbf{Z}_p,$
so elements of this subgroup can be viewed as comprising a vector space of dimension $n$ over the finite field of $p$ elements $\mathbb{F}_p$. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so
$\operatorname{Aut}(P)\cong\mathrm{GL}(n,\mathbf{F}_p),$
where $\mathrm{GL}$ is the appropriate general linear group. This is easily shown to have order
$ \left|\operatorname{Aut}(P)\right|=(p^n-1)\cdots(p^n-p^{n-1}).$
In the most general case, where the $e_i$ and $n$ are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines
$d_k=\max\{r\mid e_r = e_k\}$
and
$c_k=\min\{r\mid e_r=e_k\}$
then one has in particular $k \le d_k$, $c_k \le k$, and
$ \left|\operatorname{Aut}(P)\right| = \prod_{k=1}^n (p^{d_k}-p^{k-1}) \prod_{j=1}^n (p^{e_j})^{n-d_j} \prod_{i=1}^n (p^{e_i-1})^{n-c_i+1}. $
One can check that this yields the orders in the previous examples as special cases (see Hillar, C., & Rhea, D.).
Finitely generated abelian groups.
An abelian group A is finitely generated if it contains a finite set of elements (called "generators") $G=\{x_1, \ldots, x_n\}$ such that every element of the group is a linear combination with integer coefficients of elements of G.
Let L be a free abelian group with basis $B=\{b_1, \ldots, b_n\}.$
There is a unique group homomorphism
$p\colon L \to A,$ such that
$p(b_i) = x_i\quad \text{for } i=1,\ldots, n.$
This homomorphism is surjective, and its kernel is finitely generated (since integers form a Noetherian ring). Consider the matrix M with integer entries, such that the entries of its jth column are the coefficients of the jth generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by M. Conversely every integer matrix defines a finitely generated abelian group.
It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices. In particular, changing the generating set of A is equivalent with multiplying M on the left by a unimodular matrix (that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of M is equivalent with multiplying M on the right by a unimodular matrix.
The Smith normal form of M is a matrix
$S=UMV,$
where U and V are unimodular, and S is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries } are the first ones, and } is a divisor of } for "i" > "j". The existence and the shape of the Smith normal proves that the finitely generated abelian group A is the direct sum
$\Z^r \oplus \Z/d_{1,1}\Z \oplus \cdots \oplus \Z/d_{k,k}\Z,$
where r is the number of zero rows at the bottom of r (and also the rank of the group). This is the fundamental theorem of finitely generated abelian groups.
The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums.
Infinite abelian groups.
The simplest infinite abelian group is the infinite cyclic group $\mathbb{Z}$. Any finitely generated abelian group $A$ is isomorphic to the direct sum of $r$ copies of $\mathbb{Z}$ and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of prime power orders. Even though the decomposition is not unique, the number $r$, called the rank of $A$, and the prime powers giving the orders of finite cyclic summands are uniquely determined.
By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i.e. abelian groups $A$ in which the equation $nx = a$ admits a solution $x \in A$ for any natural number $n$ and element $a$ of $A$, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to $\mathbb{Q}$ and Prüfer groups $\mathbb{Q}_p/Z_p$ for various prime numbers $p$, and the cardinality of the set of summands of each type is uniquely determined. Moreover, if a divisible group $A$ is a subgroup of an abelian group $G$ then $A$ admits a direct complement: a subgroup $C$ of $G$ such that $G = A \oplus C$. Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is called reduced.
Two important special classes of infinite abelian groups with diametrically opposite properties are "torsion groups" and "torsion-free groups", exemplified by the groups $\mathbb{Q}/\mathbb{Z}$ (periodic) and $\mathbb{Q}$ (torsion-free).
Torsion groups.
An abelian group is called periodic or torsion, if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Prüfer theorems state that if $A$ is a periodic group, and it either has a bounded exponent, i.e., $nA = 0$ for some natural number $n$, or is countable and the $p$-heights of the elements of $A$ are finite for each $p$, then $A$ is isomorphic to a direct sum of finite cyclic groups.
The cardinality of the set of direct summands isomorphic to $\mathbb{Z}/p^m\mathbb{Z}$ in such a decomposition is an invariant of $A$. These theorems were later subsumed in the Kulikov criterion. In a different direction, Helmut Ulm found an extension of the second Prüfer theorem to countable abelian $p$-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.
Torsion-free and mixed groups.
An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively:
An abelian group that is neither periodic nor torsion-free is called mixed. If $A$ is an abelian group and $T(A)$ is its torsion subgroup, then the factor group $A/T(A)$ is torsion-free. However, in general the torsion subgroup is not a direct summand of $A$, so $A$ is "not" isomorphic to $T(A) \oplus A/T(A)$. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group $\mathbb{Z}$ of integers is torsion-free $\mathbb{Z}$-module.
Invariants and classification.
One of the most basic invariants of an infinite abelian group $A$ is its rank: the cardinality of the maximal linearly independent subset of $A$. Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of $\mathbb{Q}$ and can be completely described. More generally, a torsion-free abelian group of finite rank $r$ is a subgroup of $\mathbb{Q}_r$. On the other hand, the group of $p$-adic integers $\mathbb{Z}_p$ is a torsion-free abelian group of infinite $\mathbb{Z}$-rank and the groups $\mathbb{Z}_p^n$ with different $n$ are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.
The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by Irving Kaplansky, László Fuchs, Phillip Griffith, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in "Lecture Notes in Mathematics" for more recent findings.
Additive groups of rings.
The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are:
Relation to other mathematical topics.
Many large abelian groups possess a natural topology, which turns them into topological groups.
The collection of all abelian groups, together with the homomorphisms between them, forms the category $\textbf{Ab}$, the prototype of an abelian category.
Wanda Szmielew (1955) proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Most algebraic structures other than Boolean algebras are undecidable.
There are still many areas of current research:
Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is:
A note on typography.
Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, the lack of capitalization being a tacit acknowledgment not only of the degree to which Abel's name has been institutionalized but also of how ubiquitous in modern mathematics are the concepts introduced by him.
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Factorization algorithm
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of bits) is of the form
$\exp\left(\left((64/9)^{1/3}+o(1)\right)\left(\log n\right)^{1/3}\left(\log\log n\right)^{2/3}\right)=L_n\left[1/3,(64/9)^{1/3}\right]$
in O and L-notations. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots).
The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler rational sieve or quadratic sieve. When using such algorithms to factor a large number n, it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order "n"1/2. The size of these values is exponential in the size of n (see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. In order to achieve this speed-up, the number field sieve has to perform computations and factorizations in number fields. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.
The size of the input to the algorithm is log2 "n" or the number of bits in the binary representation of n. Any element of the order "n""c" for a constant c is exponential in log "n". The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.
Number fields.
Suppose f is a k-degree polynomial over Q (the rational numbers), and r is a complex root of f. Then, "f"("r")
0, which can be rearranged to express "r""k" as a linear combination of powers of r less than k. This equation can be used to reduce away any powers of "r" with exponent "e" ≥ "k". For example, if "f"("x")
"x"2 + 1 and r is the imaginary unit i, then "i"2 + 1
0, or "i"2
−1. This allows us to define the complex product:
$(a+bi)(c+di) = ac + (ad+bc)i + (bd)i^2 = (ac - bd) + (ad+bc)i.$
In general, this leads directly to the algebraic number field Q["r"], which can be defined as the set of complex numbers given by:
$a_{k-1}r^{k-1} + ... + a_{1}r^{1} + a_{0}r^{0}, \text{ where } a_0...,a_{k-1} \text{ in } \mathbf{Q}.$
The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of "r" with exponent "e" ≥ "k" as described above, yielding a value in the same form. To ensure that this field is actually k-dimensional and does not collapse to an even smaller field, it is sufficient that f is an irreducible polynomial over the rationals. Similarly, one may define the ring of integers OQ["r"] as the subset of Q["r"] which are roots of monic polynomials with integer coefficients. In some cases, this ring of integers is equivalent to the ring Z["r"]. However, there are many exceptions, such as for Q[√d] when d is equal to 1 modulo 4.
Method.
Two polynomials "f"("x") and "g"("x") of small degrees "d" and "e" are chosen, which have integer coefficients, which are irreducible over the rationals, and which, when interpreted mod "n", have a common integer root "m". An optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree "d" for a polynomial, consider the expansion of "n" in base "m" (allowing digits between −"m" and "m") for a number of different "m" of order "n"1/"d", and pick "f"("x") as the polynomial with the smallest coefficients and "g"("x") as "x" − "m".
Consider the number field rings Z["r"1] and Z["r"2], where "r"1 and "r"2 are roots of the polynomials "f" and "g". Since "f" is of degree "d" with integer coefficients, if "a" and "b" are integers, then so will be "b""d"·"f"("a"/"b"), which we call "r". Similarly, "s" = "b""e"·"g"("a"/"b") is an integer. The goal is to find integer values of "a" and "b" that simultaneously make "r" and "s" smooth relative to the chosen basis of primes. If "a" and "b" are small, then "r" and "s" will be small too, about the size of "m", and we have a better chance for them to be smooth at the same time. The current best-known approach for this search is lattice sieving; to get acceptable yields, it is necessary to use a large factor base.
Having enough such pairs, using Gaussian elimination, one can get products of certain "r" and of the corresponding "s" to be squares at the same time. A slightly stronger condition is needed—that they are norms of squares in our number fields, but that condition can be achieved by this method too. Each "r" is a norm of "a" − "r"1"b" and hence that the product of the corresponding factors "a" − "r"1"b" is a square in Z["r"1], with a "square root" which can be determined (as a product of known factors in Z["r"1])—it will typically be represented as an irrational algebraic number. Similarly, the product of the factors "a" − "r"2"b" is a square in Z["r"2], with a "square root" which also can be computed. It should be remarked that the use of Gaussian elimination does not give the optimal run time of the algorithm. Instead, sparse matrix solving algorithms such as Block Lanczos or Block Wiedemann are used.
Since "m" is a root of both "f" and "g" mod "n", there are homomorphisms from the rings Z["r"1] and Z["r"2] to the ring Z/"n"Z (the integers modulo "n"), which map "r"1 and "r"2 to "m", and these homomorphisms will map each "square root" (typically not represented as a rational number) into its integer representative. Now the product of the factors "a" − "mb" mod "n" can be obtained as a square in two ways—one for each homomorphism. Thus, one can find two numbers "x" and "y", with "x"2 − "y"2 divisible by "n" and again with probability at least one half we get a factor of "n" by finding the greatest common divisor of "n" and "x" − "y".
Improving polynomial choice.
The choice of polynomial can dramatically affect the time to complete the remainder of the algorithm. The method of choosing polynomials based on the expansion of n in base m shown above is suboptimal in many practical situations, leading to the development of better methods.
One such method was suggested by Murphy and Brent; they introduce a two-part score for polynomials, based on the presence of roots modulo small primes and on the average value that the polynomial takes over the sieving area.
The best reported results were achieved by the method of Thorsten Kleinjung, which allows "g"("x")
"ax" + "b", and searches over a composed of small prime factors congruent to 1 modulo 2"d" and over leading coefficients of f which are divisible by 60.
Implementations.
Some implementations focus on a certain smaller class of numbers. These are known as special number field sieve techniques, such as used in the Cunningham project.
A project called NFSNET ran from 2002 through at least 2007. It used volunteer distributed computing on the Internet.
Paul Leyland of the United Kingdom and Richard Wackerbarth of Texas were involved.
Until 2007, the gold-standard implementation was a suite of software developed and distributed by CWI in the Netherlands, which was available only under a relatively restrictive license. In 2007, Jason Papadopoulos developed a faster implementation of final processing as part of msieve, which is in the public domain. Both implementations feature the ability to be distributed among several nodes in a cluster with a sufficiently fast interconnect.
Polynomial selection is normally performed by GPL software written by Kleinjung, or by msieve, and lattice sieving by GPL software written by Franke and Kleinjung; these are distributed in GGNFS.
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Mathematical group
In mathematics, the outer automorphism group of a group, G, is the quotient, Aut("G") / Inn("G"), where Aut("G") is the automorphism group of G and Inn("G") is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out("G"). If Out("G") is trivial and G has a trivial center, then G is said to be complete.
An automorphism of a group that is not inner is called an outer automorphism. The cosets of Inn("G") with respect to outer automorphisms are then the elements of Out("G"); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group.
For example, for the alternating group, A"n", the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering A"n" as a subgroup of the symmetric group, S"n", conjugation by any odd permutation is an outer automorphism of A"n" or more precisely "represents the class of the (non-trivial) outer automorphism of A"n"", but the outer automorphism does not correspond to conjugation by any "particular" odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.
Structure.
The Schreier conjecture asserts that Out("G") is always a solvable group when G is a finite simple group. This result is now known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known.
As dual of the center.
The outer automorphism group is dual to the center in the following sense: conjugation by an element of G is an automorphism, yielding a map "σ" : "G" → Aut("G"). The kernel of the conjugation map is the center, while the cokernel is the outer automorphism group (and the image is the inner automorphism group). This can be summarized by the exact sequence:
Z("G") ↪ "G" "σ"→ Aut("G") ↠ Out("G").
Applications.
The outer automorphism group of a group acts on conjugacy classes, and accordingly on the character table. See details at character table: outer automorphisms.
Topology of surfaces.
The outer automorphism group is important in the topology of surfaces because there is a connection provided by the Dehn–Nielsen theorem: the extended mapping class group of the surface is the outer automorphism group of its fundamental group.
In finite groups.
For the outer automorphism groups of all finite simple groups see the list of finite simple groups. Sporadic simple groups and alternating groups (other than the alternating group, A6; see below) all have outer automorphism groups of order 1 or 2. The outer automorphism group of a finite simple group of Lie type is an extension of a group of "diagonal automorphisms" (cyclic except for D"n"("q"), when it has order 4), a group of "field automorphisms" (always cyclic), and a group of "graph automorphisms" (of order 1 or 2 except for D4("q"), when it is the symmetric group on 3 points). These extensions are not always semidirect products, as the case of the alternating group A6 shows; a precise criterion for this to happen was given in 2003.
In symmetric and alternating groups.
The outer automorphism group of a finite simple group in some infinite family of finite simple groups can almost always be given by a uniform formula that works for all elements of the family. There is just one exception to this: the alternating group A6 has outer automorphism group of order 4, rather than 2 as do the other simple alternating groups (given by conjugation by an odd permutation). Equivalently the symmetric group S6 is the only symmetric group with a non-trivial outer automorphism group.
$\begin{align}
n \neq 6: \operatorname{Out}(\mathrm{S}_n) & = \mathrm{C}_1 \\
n \geq 3,\ n \neq 6: \operatorname{Out}(\mathrm{A}_n) & = \mathrm{C}_2 \\
\operatorname{Out}(\mathrm{S}_6) & = \mathrm{C}_2 \\
\operatorname{Out}(\mathrm{A}_6) & = \mathrm{C}_2 \times \mathrm{C}_2
\end{align}$
Note that, in the case of "G"
A6
PSL(2, 9), the sequence 1 ⟶ "G" ⟶ Aut("G") ⟶ Out("G") ⟶ 1 does not split. A similar result holds for any PSL(2, "q"2), q odd.
In reductive algebraic groups.
Let G now be a connected reductive group over an algebraically closed field. Then any two Borel subgroups are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel subgroup. Associated to the Borel subgroup is a set of simple roots, and the outer automorphism may permute them, while preserving the structure of the associated Dynkin diagram. In this way one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out("G").
D4 has a very symmetric Dynkin diagram, which yields a large outer automorphism group of Spin(8), namely Out(Spin(8))
S3; this is called triality.
In complex and real simple Lie algebras.
The preceding interpretation of outer automorphisms as symmetries of a Dynkin diagram follows from the general fact, that for a complex or real simple Lie algebra, 𝔤, the automorphism group Aut("𝔤") is a semidirect product of Inn("𝔤") and Out("𝔤"); i.e., the short exact sequence
1 ⟶ Inn("𝔤") ⟶ Aut("𝔤") ⟶ Out("𝔤") ⟶ 1
splits. In the complex simple case, this is a classical result, whereas for real simple Lie algebras, this fact has been proven as recently as 2010.
Word play.
The term "outer automorphism" lends itself to word play: the term "outermorphism" is sometimes used for "outer automorphism", and a particular geometry on which Out("F""n") acts is called "outer space".
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Nonabelian group of order 120
In mathematics, the binary icosahedral group 2"I" or ⟨2,3,5⟩ is a certain nonabelian group of order 120.
It is an extension of the icosahedral group "I" or (2,3,5) of order 60 by the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism
$\operatorname{Spin}(3) \to \operatorname{SO}(3)\,$
of the special orthogonal group by the spin group. It follows that the binary icosahedral group is a discrete subgroup of Spin(3) of order 120.
It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O(3).
The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism $\operatorname{Spin}(3) \cong \operatorname{Sp}(1)$ where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Elements.
Explicitly, the binary icosahedral group is given as the union of all even permutations of the following vectors:
Here $\phi = \frac{1 + \sqrt{5}} {2}$ is the golden ratio.
In total there are 120 elements, namely the unit icosians. They all have unit magnitude and therefore lie in the unit quaternion group Sp(1).
The 120 elements in 4-dimensional space match the 120 vertices the 600-cell, a regular 4-polytope.
Properties.
Central extension.
The binary icosahedral group, denoted by 2"I", is the universal perfect central extension of the icosahedral group, and thus is quasisimple: it is a perfect central extension of a simple group.
Explicitly, it fits into the short exact sequence
$1\to\{\pm 1\}\to 2I\to I \to 1.\,$
This sequence does not split, meaning that 2"I" is "not" a semidirect product of { ±1 } by "I". In fact, there is no subgroup of 2"I" isomorphic to "I".
The center of 2"I" is the subgroup { ±1 }, so that the inner automorphism group is isomorphic to "I". The full automorphism group is isomorphic to "S"5 (the symmetric group on 5 letters), just as for $I\cong A_5$ - any automorphism of 2"I" fixes the non-trivial element of the center ($-1$), hence descends to an automorphism of "I," and conversely, any automorphism of "I" lifts to an automorphism of 2"I," since the lift of generators of "I" are generators of 2"I" (different lifts give the same automorphism).
Superperfect.
The binary icosahedral group is perfect, meaning that it is equal to its commutator subgroup. In fact, 2"I" is the unique perfect group of order 120. It follows that 2"I" is not solvable.
Further, the binary icosahedral group is superperfect, meaning abstractly that its first two group homology groups vanish: $H_1(2I;\mathbf{Z})\cong H_2(2I;\mathbf{Z})\cong 0.$ Concretely, this means that its abelianization is trivial (it has no non-trivial abelian quotients) and that its Schur multiplier is trivial (it has no non-trivial perfect central extensions). In fact, the binary icosahedral group is the smallest (non-trivial) superperfect group.
The binary icosahedral group is not acyclic, however, as H"n"(2"I",Z) is cyclic of order 120 for "n" = 4"k"+3, and trivial for "n" > 0 otherwise, .
Isomorphisms.
Concretely, the binary icosahedral group is a subgroup of Spin(3), and covers the icosahedral group, which is a subgroup of SO(3). Abstractly, the icosahedral group is isomorphic to the symmetries of the 4-simplex, which is a subgroup of SO(4), and the binary icosahedral group is isomorphic to the double cover of this in Spin(4). Note that the symmetric group $S_5$ "does" have a 4-dimensional representation (its usual lowest-dimensional irreducible representation as the full symmetries of the $(n-1)$-simplex), and that the full symmetries of the 4-simplex are thus $S_5,$ not the full icosahedral group (these are two different groups of order 120).
The binary icosahedral group can be considered as the double cover of the alternating group $A_5,$ denoted $2\cdot A_5 \cong 2I;$ this isomorphism covers the isomorphism of the icosahedral group with the alternating group $A_5 \cong I,$.
Just as $I$ is a discrete subgroup of $\mathrm{SO}(3)$, $2I$ is a discrete subgroup of the double over of $\mathrm{SO}(3)$, namely $\mathrm{Spin}(3) \cong \mathrm{SU}(2)$. The 2-1 homomorphism from $\mathrm{Spin}(3)$ to $\mathrm{SO}(3)$ then restricts to the 2-1 homomorphism from $2I$ to $I$.
One can show that the binary icosahedral group is isomorphic to the special linear group SL(2,5) — the group of all 2×2 matrices over the finite field F5 with unit determinant; this covers the exceptional isomorphism of $I\cong A_5$ with the projective special linear group PSL(2,5).
Note also the exceptional isomorphism $PGL(2,5) \cong S_5,$ which is a different group of order 120, with the commutative square of SL, GL, PSL, PGL being isomorphic to a commutative square of $2\cdot A_5, 2\cdot S_5, A_5, S_5,$ which are isomorphic to subgroups of the commutative square of Spin(4), Pin(4), SO(4), O(4).
Presentation.
The group 2"I" has a presentation given by
$\langle r,s,t \mid r^2 = s^3 = t^5 = rst \rangle$
or equivalently,
$\langle s,t \mid (st)^2 = s^3 = t^5 \rangle.$
Generators with these relations are given by
$s = \tfrac{1}{2}(1+i+j+k) \qquad t = \tfrac{1}{2}(\varphi+\varphi^{-1}i+j).$
Subgroups.
The only proper normal subgroup of 2"I" is the center { ±1 }.
By the third isomorphism theorem, there is a Galois connection between subgroups of 2"I" and subgroups of "I", where the closure operator on subgroups of 2"I" is multiplication by { ±1 }.
$-1$ is the only element of order 2, hence it is contained in all subgroups of even order: thus every subgroup of 2"I" is either of odd order or is the preimage of a subgroup of "I".
Besides the cyclic groups generated by the various elements (which can have odd order), the only other subgroups of 2"I" (up to conjugation) are:
Relation to 4-dimensional symmetry groups.
The 4-dimensional analog of the icosahedral symmetry group "I"h is the symmetry group of the 600-cell (also that of its dual, the 120-cell). Just as the former is the Coxeter group of type "H"3, the latter is the Coxeter group of type "H"4, also denoted [3,3,5]. Its rotational subgroup, denoted [3,3,5]+ is a group of order 7200 living in SO(4). SO(4) has a double cover called Spin(4) in much the same way that Spin(3) is the double cover of SO(3). Similar to the isomorphism Spin(3) = Sp(1), the group Spin(4) is isomorphic to Sp(1) × Sp(1).
The preimage of [3,3,5]+ in Spin(4) (a four-dimensional analogue of 2"I") is precisely the product group 2"I" × 2"I" of order 14400. The rotational symmetry group of the 600-cell is then
[3,3,5]+ = ( 2"I" × 2"I" ) / { ±1 }.
Various other 4-dimensional symmetry groups can be constructed from 2"I". For details, see (Conway and Smith, 2003).
Applications.
The coset space Spin(3) / 2"I" = "S"3 / 2"I" is a spherical 3-manifold called the Poincaré homology sphere. It is an example of a homology sphere, i.e. a 3-manifold whose homology groups are identical to those of a 3-sphere. The fundamental group of the Poincaré sphere is isomorphic to the binary icosahedral group, as the Poincaré sphere is the quotient of a 3-sphere by the binary icosahedral group.
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In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic ring structure is a more recent development, due to Solomon (1967).
Formal definition.
Given a finite group "G", the generators of its Burnside ring Ω("G") are the formal sums of isomorphism classes of finite "G"-sets. For the ring structure, addition is given by disjoint union of "G"-sets and multiplication by their Cartesian product.
The Burnside ring is a free Z-module, whose generators are the (isomorphism classes of) orbit types of "G".
If "G" acts on a finite set "X", then one can write <math display="inline">X = \bigcup_i X_i$ (disjoint union), where each "X""i" is a single "G"-orbit. Choosing any element "x""i" in "X"i creates an isomorphism "G"/"G""i" → "X""i", where "Gi" is the stabilizer (isotropy) subgroup of "G" at "x""i". A different choice of representative "y""i" in "X""i" gives a conjugate subgroup to "G""i" as stabilizer. This shows that the generators of Ω("G") as a Z-module are the orbits "G"/"H" as "H" ranges over conjugacy classes of subgroups of "G".
In other words, a typical element of Ω("G") is
<math display="block"> \sum_{i=1}^N a_i [G/G_i],$
where "a""i" in Z and "G"1, "G"2, ..., "G""N" are representatives of the conjugacy classes of subgroups of "G".
Marks.
Much as character theory simplifies working with group representations, marks simplify working with permutation representations and the Burnside ring.
If "G" acts on "X", and "H" ≤ "G" ("H" is a subgroup of "G"), then the mark of "H" on "X" is the number of elements of "X" that are fixed by every element of "H": $m_X(H) = \left|X^H\right|$, where
$X^H = \{ x\in X \mid h\cdot x = x, \forall h\in H\}.$
If "H" and "K" are conjugate subgroups, then "m""X"("H") = "m""X"("K") for any finite "G"-set "X"; indeed, if "K" = "gHg"−1 then "X""K" = "g" · "X""H".
It is also easy to see that for each "H" ≤ "G", the map "Ω"("G") → Z : "X" ↦ "m""X"("H") is a homomorphism. This means that to know the marks of "G", it is sufficient to evaluate them on the generators of "Ω"("G"), "viz." the orbits "G"/"H".
For each pair of subgroups "H","K" ≤ "G" define
$m(K, H) = \left|[G/K]^H\right| = \# \left\{ gK \in G/K \mid HgK=gK \right\}.$
This is "m""X"("H") for "X" = "G"/"K". The condition "HgK" = "gK" is equivalent to "g"−1"Hg" ≤ "K", so if "H" is not conjugate to a subgroup of "K" then "m"("K", "H") = 0.
To record all possible marks, one forms a table, Burnside's Table of Marks, as follows: Let "G"1 (= trivial subgroup), "G"2, ..., "G""N" = "G" be representatives of the "N" conjugacy classes of subgroups of "G", ordered in such a way that whenever "G""i" is conjugate to a subgroup of "G""j", then "i" ≤ "j". Now define the "N" × "N" table (square matrix) whose ("i", "j")th entry is "m"("G""i", "G""j"). This matrix is lower triangular, and the elements on the diagonal are non-zero so it is invertible.
It follows that if "X" is a "G"-set, and u its row vector of marks, so "u""i" = "m""X"("G""i"), then "X" decomposes as a disjoint union of "a""i" copies of the orbit of type "G""i", where the vector a satisfies,
aM" = u",
where "M" is the matrix of the table of marks. This theorem is due to .
Examples.
The table of marks for the cyclic group of order 6:
The table of marks for the symmetric group "S3":
The dots in the two tables are all zeros, merely emphasizing the fact that the tables are lower-triangular.
The fact that the last row is all 1s is because ["G"/"G"] is a single point. The diagonal terms are "m"("H", "H") = | "N""G"("H")/"H" |. The numbers in the first column show the degree of the representation.
The ring structure of "Ω"("G") can be deduced from these tables: the generators of the ring (as a Z-module) are the rows of the table, and the product of two generators has mark given by the product of the marks (so component-wise multiplication of row vectors), which can then be decomposed as a linear combination of all the rows. For example, with "S"3,
$[G/\mathbf{Z}_2]\cdot[G/\mathbf{Z}_3] = [G/1],$
as (3, 1, 0, 0).(2, 0, 2, 0) = (6, 0, 0, 0).
Permutation representations.
Associated to any finite set "X" is a vector space "V = VX", which is the vector space with the elements of "X" as the basis (using any specified field). An action of a finite group "G" on "X" induces a linear action on "V", called a permutation representation. The set of all finite-dimensional representations of "G" has the structure of a ring, the representation ring, denoted "R(G)".
For a given "G"-set "X", the character of the associated representation is
$\chi(g) = m_X(\langle g\rangle)$
where $\langle g\rangle$ is the cyclic group generated by $g$.
The resulting map
$\beta : \Omega(G) \longrightarrow R(G) $
taking a "G"-set to the corresponding representation is in general neither injective nor surjective.
The simplest example showing that β is not in general injective is for "G = S3" (see table above), and is given by
$\beta(2[S_3/\mathbf{Z}_2] + [S_3/\mathbf{Z}_3]) = \beta([S_3] + 2[S_3/S_3]).$
Extensions.
The Burnside ring for compact groups is described in .
The Segal conjecture relates the Burnside ring to homotopy.
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In the area of modern algebra known as group theory, the Mathieu group "M12" is a sporadic simple group of order
12 · 11 · 10 · 9 · 8 = 26 · 33 · 5 · 11 = 95040.
History and properties.
"M12" is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a sharply 5-transitive permutation group on 12 objects. showed that the Schur multiplier of M12 has order 2 (correcting a mistake in where they incorrectly claimed it has order 1).
The double cover had been implicitly found earlier by , who showed that M12 is a subgroup of the projective linear group of dimension 6 over the finite field with 3 elements.
The outer automorphism group has order 2, and the full automorphism group M12.2 is contained in M24 as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M12 swapping the two dodecads.
Representations.
calculated the complex character table of M12.
M12 has a strictly 5-transitive permutation representation on 12 points, whose point stabilizer is the Mathieu group M11. Identifying the 12 points with the projective line over the field of 11 elements, M12 is generated by the permutations of PSL2(11) together with the permutation (2,10)(3,4)(5,9)(6,7). This permutation representation preserves a Steiner system S(5,6,12) of 132 special hexads, such that each pentad is contained in exactly 1 special hexad, and the hexads are the supports of the weight 6 codewords of the extended ternary Golay code. In fact M12 has two inequivalent actions on 12 points, exchanged by an outer automorphism; these are analogous to the two inequivalent actions of the symmetric group "S"6 on 6 points.
The double cover 2.M12 is the automorphism group of the extended ternary Golay code, a dimension 6 length 12 code over the field of order 3 of minimum weight 6. In particular the double cover has an irreducible 6-dimensional representation over the field of 3 elements.
The double cover 2.M12 is the automorphism group of any 12×12 Hadamard matrix.
M12 centralizes an element of order 11 in the monster group, as a result of which it acts naturally on a vertex algebra over the field with 11 elements, given as the Tate cohomology of the monster vertex algebra.
Maximal subgroups.
There are 11 conjugacy classes of maximal subgroups of M12, 6 occurring in automorphic pairs, as follows:
Isomorphic to the affine group on the space C3 x C3.
Centralizer of a sextuple transposition
Centralizer of a quadruple transposition
Conjugacy classes.
The cycle shape of an element and its conjugate under an outer automorphism are related in the following way: the union of the two cycle shapes is balanced, in other words invariant under changing each "n"-cycle to an "N"/"n" cycle for some integer "N".
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In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group $G$, the holomorph of $G$ denoted $\operatorname{Hol}(G)$ can be described as a semidirect product or as a permutation group.
Hol("G") as a semidirect product.
If $\operatorname{Aut}(G)$ is the automorphism group of $G$ then
$\operatorname{Hol}(G)=G\rtimes \operatorname{Aut}(G)$
where the multiplication is given by
$(g,\alpha)(h,\beta)=(g\alpha(h),\alpha\beta).$ [Eq. 1]
Typically, a semidirect product is given in the form $G\rtimes_{\phi}A$ where $G$ and $A$ are groups and $\phi:A\rightarrow \operatorname{Aut}(G)$ is a homomorphism and where the multiplication of elements in the semidirect product is given as
$(g,a)(h,b)=(g\phi(a)(h),ab)$
which is well defined, since $\phi(a)\in \operatorname{Aut}(G)$ and therefore $\phi(a)(h)\in G$.
For the holomorph, $A=\operatorname{Aut}(G)$ and $\phi$ is the identity map, as such we suppress writing $\phi$ explicitly in the multiplication given in [Eq. 1] above.
For example,
$(x^{i_1},\sigma^{j_1})(x^{i_2},\sigma^{j_2}) = (x^{i_1+i_22^{^{j_1}}},\sigma^{j_1+j_2})$ where the exponents of $x$ are taken mod 3 and those of $\sigma$ mod 2.
Observe, for example
$(x,\sigma)(x^2,\sigma)=(x^{1+2\cdot2},\sigma^2)=(x^2,1)$
and this group is not abelian, as $(x^2,\sigma)(x,\sigma)=(x,1)$, so that $\operatorname{Hol}(C_3)$ is a non-abelian group of order 6, which, by basic group theory, must be isomorphic to the symmetric group $S_3$.
Hol("G") as a permutation group.
A group "G" acts naturally on itself by left and right multiplication, each giving rise to a homomorphism from "G" into the symmetric group on the underlying set of "G". One homomorphism is defined as "λ": "G" → Sym("G"), $\lambda_g$("h") = "g"·"h". That is, "g" is mapped to the permutation obtained by left-multiplying each element of "G" by "g". Similarly, a second homomorphism "ρ": "G" → Sym("G") is defined by $\rho_g$("h") = "h"·"g"−1, where the inverse ensures that $\rho_{gh}$("k") = $\rho_g$($\rho_h$("k")). These homomorphisms are called the left and right regular representations of "G". Each homomorphism is injective, a fact referred to as Cayley's theorem.
For example, if "G" = "C"3 = {1, "x", "x"2 } is a cyclic group of order three, then
so "λ"("x") takes (1, "x", "x"2) to ("x", "x"2, 1).
The image of "λ" is a subgroup of Sym("G") isomorphic to "G", and its normalizer in Sym("G") is defined to be the holomorph "N" of "G".
For each "n" in "N" and "g" in "G", there is an "h" in "G" such that "n"·$\lambda_g$ = $\lambda_h$·"n". If an element "n" of the holomorph fixes the identity of "G", then for 1 in "G", ("n"·$\lambda_g$)(1) = ($\lambda_h$·"n")(1), but the left hand side is "n"("g"), and the right side is "h". In other words, if "n" in "N" fixes the identity of "G", then for every "g" in "G", "n"·$\lambda_g$ = $\lambda_{n(g)}$·"n". If "g", "h" are elements of "G", and "n" is an element of "N" fixing the identity of "G", then applying this equality twice to "n"·$\lambda_g$·$\lambda_h$ and once to the (equivalent) expression "n"·$\lambda_{gh}$ gives that "n"("g")·"n"("h") = "n"("g"·"h"). That is, every element of "N" that fixes the identity of "G" is in fact an automorphism of "G". Such an "n" normalizes $\lambda_G$, and the only $\lambda_g$ that fixes the identity is "λ"(1). Setting "A" to be the stabilizer of the identity, the subgroup generated by "A" and $\lambda_G$ is semidirect product with normal subgroup $\lambda_G$ and complement "A". Since $\lambda_G$ is transitive, the subgroup generated by $\lambda_G$ and the point stabilizer "A" is all of "N", which shows the holomorph as a permutation group is isomorphic to the holomorph as semidirect product.
It is useful, but not directly relevant, that the centralizer of $\lambda_G$ in Sym("G") is $\rho_G$, their intersection is $\rho_{Z(G)}=\lambda_{Z(G)}$, where Z("G") is the center of "G", and that "A" is a common complement to both of these normal subgroups of "N".
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In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms.
Dedicated to the discrete logarithm in $(\mathbb{Z}/q\mathbb{Z})^*$ where $q$ is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic curves. The algorithm collects relations among the discrete logarithms of small primes, computes them by a linear algebra procedure and finally expresses the desired discrete logarithm with respect to the discrete logarithms of small primes.
Description.
Roughly speaking, the discrete log problem asks us to find an "x" such that $g^x \equiv h \pmod{n}$, where "g", "h", and the modulus "n" are given.
The algorithm (described in detail below) applies to the group $(\mathbb{Z}/q\mathbb{Z})^*$ where "q" is prime. It requires a "factor base" as input. This "factor base" is usually chosen to be the number −1 and the first "r" primes starting with 2. From the point of view of efficiency, we want this factor base to be small, but in order to solve the discrete log for a large group we require the "factor base" to be (relatively) large. In practical implementations of the algorithm, those conflicting objectives are compromised one way or another.
The algorithm is performed in three stages. The first two stages depend only on the generator "g" and prime modulus "q", and find the discrete logarithms of a "factor base" of "r" small primes. The third stage finds the discrete log of the desired number "h" in terms of the discrete logs of the factor base.
The first stage consists of searching for a set of "r" linearly independent "relations" between the factor base and power of the generator "g". Each relation contributes one equation to a system of linear equations in "r" unknowns, namely the discrete logarithms of the "r" primes in the factor base. This stage is embarrassingly parallel and easy to divide among many computers.
The second stage solves the system of linear equations to compute the discrete logs of the factor base. A system of hundreds of thousands or millions of equations is a significant computation requiring large amounts of memory, and it is "not" embarrassingly parallel, so a supercomputer is typically used. This was considered a minor step compared to the others for smaller discrete log computations. However, larger discrete logarithm records were made possible only by shifting the work away from the linear algebra and onto the sieve (i.e., increasing the number of equations while reducing the number of variables).
The third stage searches for a power "s" of the generator "g" which, when multiplied by the argument "h", may be factored in terms of the factor base "gsh" = (−1)"f"0 2"f"1 3"f"2···"p""r""f""r".
Finally, in an operation too simple to really be called a fourth stage, the results of the second and third stages can be rearranged by simple algebraic manipulation to work out the desired discrete logarithm "x" = "f"0log"g"(−1) + "f"1log"g"2 + "f"2log"g"3 + ··· + "f""r"log"g""pr" − "s".
The first and third stages are both embarrassingly parallel, and in fact the third stage does not depend on the results of the first two stages, so it may be done in parallel with them.
The choice of the factor base size "r" is critical, and the details are too intricate to explain here. The larger the factor base, the easier it is to find relations in stage 1, and the easier it is to complete stage 3, but the more relations you need before you can proceed to stage 2, and the more difficult stage 2 is. The relative availability of computers suitable for the different types of computation required for stages 1 and 2 is also important.
Applications in other groups.
The lack of the notion of "prime elements" in the group of points on elliptic curves makes it impossible to find an efficient "factor base" to run index calculus method as presented here in these groups. Therefore this algorithm is incapable of solving discrete logarithms efficiently in elliptic curve groups. However: For special kinds of curves (so called supersingular elliptic curves) there are specialized algorithms for solving the problem faster than with generic methods. While the use of these special curves can easily be avoided, in 2009 it has been proven that for certain fields the discrete logarithm problem in the group of points on "general" elliptic curves over these fields can be solved faster than with generic methods. The algorithms are indeed adaptations of the index calculus method.
The algorithm.
Input: Discrete logarithm generator $g$, modulus $q$ and argument $h$. Factor base $\{-1, 2, 3, 5, 7, 11, \ldots, p_r\}$, of length $r+1$.
Output: $x$ such that $g^x=h \mod q$.
Complexity.
Assuming an optimal selection of the factor base, the expected running time (using L-notation) of the index-calculus algorithm can be stated as
$L_n[1/2,\sqrt{2}+o(1)] $.
History.
The basic idea of the algorithm is due to Western and Miller (1968), which ultimately relies on ideas from Kraitchik (1922). The first practical implementations followed the 1976 introduction of the Diffie-Hellman cryptosystem which relies on the discrete logarithm. Merkle's Stanford University dissertation (1979) was credited by Pohlig (1977) and Hellman and Reyneri (1983), who also made improvements to the implementation. Adleman optimized the algorithm and presented it in the present form.
The Index Calculus family.
Index Calculus inspired a large family of algorithms. In finite fields $\mathbb{F}_{q} $ with $q=p^n$ for some prime $p$, the state-of-art algorithms are
the Number Field Sieve for Discrete Logarithms, <math display="inline"> L_{q}\left[1/3,\sqrt[3]{64/9}\,\right]$, when $ p $ is large compared to $q$, the function field sieve, <math display="inline">L_q\left[1/3,\sqrt[3]{32/9}\,\right]$, and Joux, $L_{q}\left[1/4+\varepsilon,c\right] $ for $c>0$, when $p$ is small compared to $q $ and the Number Field Sieve in High Degree, $L_q[1/3,c]$ for $c>0$ when $p $ is middle-sided. Discrete logarithm in some families of elliptic curves can be solved in time $L_q\left[1/3,c\right]$ for $ c>0$, but the general case remains exponential.
Notes.
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In finite group theory, Jordan's theorem states that if a primitive permutation group "G" is a subgroup of the symmetric group "S""n" and contains a "p"-cycle for some prime number "p" < "n" − 2, then "G" is either the whole symmetric group "S""n" or the alternating group "A""n". It was first proved by Camille Jordan.
The statement can be generalized to the case that "p" is a prime power.
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In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the automorphism group. Some facts:
For infinite groups, an example of a class automorphism that is not inner is the following: take the finitary symmetric group on countably many elements and consider conjugation by an infinitary permutation. This conjugation defines an outer automorphism on the group of finitary permutations. However, for any specific finitary permutation, we can find a finitary permutation whose conjugation has the same effect as this infinitary permutation. This is essentially because the infinitary permutation takes permutations of finite supports to permutations of finite support.
For finite groups, the classical example is a group of order 32 obtained as the semidirect product of the cyclic ring on 8 elements, by its group of units acting via multiplication. Finding a class automorphism in the stability group that is not inner boils down to finding a cocycle for the action that is locally a coboundary but is not a global coboundary.
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Operation in group theory
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as "semidirect products".
For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension).
Inner semidirect product definitions.
Given a group "G" with identity element "e", a subgroup "H", and a normal subgroup "N" ◁ "G", the following statements are equivalent:
$1 \to N \to G \to H \to 1$
of groups (which is also known as group extension of $H$ by $N$).
If any of these statements holds (and hence all of them hold, by their equivalence), we say "G" is the semidirect product of "N" and "H", written
$G = N \rtimes H$ or $G = H \ltimes N,$
or that "G" "splits" over "N"; one also says that "G" is a semidirect product of "H" acting on "N", or even a semidirect product of "H" and "N". To avoid ambiguity, it is advisable to specify which is the normal subgroup.
If $G = H \ltimes N$, then there is a group homomorphism $\varphi\colon H\rightarrow \mathrm{Aut} (N)$ given by $\varphi_h(n)=hnh^{-1}$, and for $g=hn,g'=h'n'$, we have $gg'=hnh'n'=hh'h'^{-1}nh'n'=hh'\varphi_(n)n' = h^* n^*$.
Inner and outer semidirect products.
Let us first consider the inner semidirect product. In this case, for a group $G$, consider its normal subgroup "N" and the subgroup "H" (not necessarily normal). Assume that the
conditions on the list above hold. Let $\operatorname{Aut}(N)$ denote the group of all automorphisms of "N", which is a group under composition. Construct a group homomorphism $\varphi \colon H \to \operatorname{Aut}(N)$ defined by conjugation,
$\varphi_h(n) = hnh^{-1}$, for all "h" in "H" and "n" in "N".
In this way we can construct a group $G'=(N,H)$ with group operation defined as
$ (n_1, h_1) \cdot (n_2, h_2) = (n_1 \varphi_{h_1}(n_2),\, h_1 h_2)$ for "n"1, "n"2 in "N" and "h"1, "h"2 in "H".
The subgroups "N" and "H" determine "G" up to isomorphism, as we will show later. In this way, we can construct the group "G" from its subgroups. This kind of construction is called an inner semidirect product (also known as internal semidirect product).
Let us now consider the outer semidirect product. Given any two groups "N" and "H" and a group homomorphism "φ": "H" → Aut("N"), we can construct a new group "N" ⋊"φ" "H", called the outer semidirect product of "N" and "H" with respect to "φ", defined as follows:
This defines a group in which the identity element is ("e""N", "eH") and the inverse of the element ("n", "h") is ("φ""h"−1("n"−1), "h"−1). Pairs ("n", "eH") form a normal subgroup isomorphic to "N", while pairs ("eN", "h") form a subgroup isomorphic to "H". The full group is a semidirect product of those two subgroups in the sense given earlier.
Conversely, suppose that we are given a group "G" with a normal subgroup "N" and a subgroup "H", such that every element "g" of "G" may be written uniquely in the form "g
nh" where "n" lies in "N" and "h" lies in "H". Let "φ": "H" → Aut("N") be the homomorphism (written "φ"("h")
"φ""h") given by
$\varphi_h(n) = hnh^{-1}$
for all "n" ∈ "N", "h" ∈ "H".
Then "G" is isomorphic to the semidirect product "N" ⋊"φ" "H". The isomorphism "λ": "G" → "N" ⋊"φ" "H" is well defined
by "λ"("a")
"λ"("nh")
("n, h") due to the uniqueness of the decomposition "a"
"nh".
In "G", we have
$(n_1 h_1)(n_2 h_2) = n_1 h_1 n_2(h_1^{-1}h_1) h_2 =
(n_1 \varphi_{h_1}(n_2))(h_1 h_2)$
Thus, for "a"
"n"1"h"1 and "b"
"n"2"h"2 we obtain
$\begin{align}
\lambda(ab) & = \lambda(n_1 h_1 n_2 h_2) = \lambda(n_1 \varphi_{h_1} (n_2) h_1 h_2) = (n_1 \varphi_{h_1} (n_2), h_1 h_2) = (n_1, h_1) \bullet (n_2, h_2) \\[5pt]
& = \lambda(n_1 h_1) \bullet \lambda(n_2 h_2) = \lambda(a) \bullet \lambda(b),
\end{align}$
which proves that "λ" is a homomorphism. Since "λ" is obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule in "N" ⋊"φ" "H".
The direct product is a special case of the semidirect product. To see this, let "φ" be the trivial homomorphism (i.e., sending every element of "H" to the identity automorphism of "N") then "N" ⋊"φ" "H" is the direct product "N" × "H".
A version of the splitting lemma for groups states that a group "G" is isomorphic to a semidirect product of the two groups "N" and "H" if and only if there exists a short exact sequence
$ 1 \longrightarrow N \,\overset{\beta}{\longrightarrow}\, G \,\overset{\alpha}{\longrightarrow}\, H \longrightarrow 1$
and a group homomorphism "γ": "H" → "G" such that "α" ∘ "γ"
id"H", the identity map on "H". In this case, "φ": "H" → Aut("N") is given by "φ"("h")
"φ""h", where
$\varphi_h(n) = \beta^{-1}(\gamma(h)\beta(n)\gamma(h^{-1})).$
Examples.
Dihedral group.
The dihedral group D2"n" with 2"n" elements is isomorphic to a semidirect product of the cyclic groups C"n" and C2. Here, the non-identity element of C2 acts on C"n" by inverting elements; this is an automorphism since C"n" is abelian. The presentation for this group is:
$\langle a,\;b \mid a^2 = e,\; b^n = e,\; aba^{-1} = b^{-1}\rangle.$
Cyclic groups.
More generally, a semidirect product of any two cyclic groups C"m" with generator "a" and C"n" with generator "b" is given by one extra relation, "aba"−1
"bk", with "k" and "n" coprime, and $k^m\equiv 1 \pmod{n}$; that is, the presentation:
$\langle a,\;b \mid a^m = e,\;b^n = e,\;aba^{-1} = b^k\rangle.$
If "r" and "m" are coprime, "ar" is a generator of C"m" and "arba−r"
"bkr", hence the presentation:
$\langle a,\;b \mid a^m = e,\;b^n = e,\;aba^{-1} = b^{k^{r}}\rangle$
gives a group isomorphic to the previous one.
Holomorph of a group.
One canonical example of a group expressed as a semi-direct product is the holomorph of a group. This is defined as$\operatorname{Hol}(G)=G\rtimes \operatorname{Aut}(G)$where $\text{Aut}(G)$ is the automorphism group of a group $G$ and the structure map $\varphi$ comes from the right action of $\text{Aut}(G)$ on $G$. In terms of multiplying elements, this gives the group structure$(g,\alpha)(h,\beta)=(g(\varphi(\alpha)\cdot h),\alpha\beta).$
Fundamental group of the Klein bottle.
The fundamental group of the Klein bottle can be presented in the form
$\langle a,\;b \mid aba^{-1} = b^{-1}\rangle.$
and is therefore a semidirect product of the group of integers, $\mathbb{Z}$, with $\mathbb{Z}$. The corresponding homomorphism "φ": $\mathbb{Z}$ → Aut($\mathbb{Z}$) is given by "φ"("h")("n")
(−1)"h""n".
Upper triangular matrices.
The group $\mathbb{T}_n$ of upper triangular matrices with non-zero determinant, that is with non-zero entries on the diagonal, has a decomposition into the semidirect product
$\mathbb{T}_n \cong \mathbb{U}_n \rtimes \mathbb{D}_n$ where $\mathbb{U}_n$ is the subgroup of matrices with only $1$'s on the diagonal, which is called the upper unitriangular matrix group, and $\mathbb{D}_n$ is the subgroup of diagonal matrices.
The group action of $\mathbb{D}_n$ on $\mathbb{U}_n$ is induced by matrix multiplication. If we set
x_1 & 0 & \cdots & 0 \\
0 & x_2 & \cdots & 0 \\
\vdots & \vdots & & \vdots \\
0 & 0 & \cdots & x_n
\end{bmatrix}$
1 & a_{12} & a_{13} & \cdots & a_{1n} \\
0 & 1 & a_{23} & \cdots & a_{2n} \\
\vdots & \vdots & \vdots & & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{bmatrix}$
then their matrix product is
$AB =
x_1 & x_1a_{12} & x_1a_{13} & \cdots & x_1a_{1n} \\
0 & x_2 & x_2a_{23} & \cdots & x_2a_{2n} \\
\vdots & \vdots & \vdots & & \vdots \\
0 & 0 & 0 & \cdots & x_n
\end{bmatrix}.$
This gives the induced group action $m:\mathbb{D}_n\times \mathbb{U}_n \to \mathbb{U}_n$
$m(A,B) = \begin{bmatrix}
1 & x_1a_{12} & x_1a_{13} & \cdots & x_1a_{1n} \\
0 & 1 & x_2a_{23} & \cdots & x_2a_{2n} \\
\vdots & \vdots & \vdots & & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{bmatrix}.$
A matrix in $\mathbb{T}_n$ can be represented by matrices in $\mathbb{U}_n$ and $\mathbb{D}_n$. Hence $\mathbb{T}_n \cong \mathbb{U}_n \rtimes \mathbb{D}_n$.
Group of isometries on the plane.
The Euclidean group of all rigid motions (isometries) of the plane (maps "f": $\mathbb{R}$2 → $\mathbb{R}$2 such that the Euclidean distance between "x" and "y" equals the distance between "f"("x") and "f"("y") for all "x" and "y" in $\mathbb{R}^2$) is isomorphic to a semidirect product of the abelian group $\mathbb{R}^2$ (which describes translations) and the group O(2) of orthogonal 2 × 2 matrices (which describes rotations and reflections that keep the origin fixed). Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the conjugate of the original translation). This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and O(2), and that the corresponding homomorphism "φ": O(2) → Aut($\mathbb{R}$2) is given by matrix multiplication: "φ"("h")("n")
"hn".
Orthogonal group O(n).
The orthogonal group O("n") of all orthogonal real "n" × "n" matrices (intuitively the set of all rotations and reflections of "n"-dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the group SO("n") (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of "n"-dimensional space) and C2. If we represent C2 as the multiplicative group of matrices {"I", "R"}, where "R" is a reflection of "n"-dimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant –1 representing an involution), then "φ": C2 → Aut(SO("n")) is given by "φ"("H")("N")
"HNH"−1 for all "H" in C2 and "N" in SO("n"). In the non-trivial case ("H" is not the identity) this means that "φ"("H") is conjugation of operations by the reflection (in 3-dimensional space a rotation axis and the direction of rotation are replaced by their "mirror image").
Semi-linear transformations.
The group of semilinear transformations on a vector space "V" over a field $\mathbb{K}$, often denoted ΓL("V"), is isomorphic to a semidirect product of the linear group GL("V") (a normal subgroup of ΓL("V")), and the automorphism group of $\mathbb{K}$.
Crystallographic groups.
In crystallography, the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group is symmorphic. Non-symmorphic space groups have point groups that are not even contained as subset of the space group, which is responsible for much of the complication in their analysis.
Non-examples.
Of course, no simple group can be expressed as a semi-direct product (because they do not have nontrivial normal subgroups), but there are a few common counterexamples of groups containing a non-trivial normal subgroup that nonetheless cannot be expressed as a semi-direct product. Note that although not every group $G$ can be expressed as a split extension of $H$ by $A$, it turns out that such a group can be embedded into the wreath product $A\wr H$ by the universal embedding theorem.
Z4.
The cyclic group $\mathbb{Z}_4$ is not a simple group since it has a subgroup of order 2, namely $\{0,2\} \cong \mathbb{Z}_2$ is a subgroup and their quotient is $\mathbb{Z}_2$, so there's an extension$0 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 0$If the extension was split, then the group $G$ in$0 \to \mathbb{Z}_2 \to G \to \mathbb{Z}_2 \to 0$would be isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$.
Q8.
The group of the eight quaternions $\{\pm 1,\pm i,\pm j,\pm k\}$ where $ijk = -1$ and $i^2 = j^2 = k^2 = -1$, is another example of a group which has non-trivial normal subgroups yet is still not split. For example, the subgroup generated by $i$ is isomorphic to $\mathbb{Z}_4$ and is normal. It also has a subgroup of order $2$ generated by $-1$. This would mean $Q_8$ would have to be a split extension in the following "hypothetical" exact sequence of groups: $0 \to \mathbb{Z}_4 \to Q_8 \to \mathbb{Z}_2 \to 0$, but such an exact sequence does not exist. This can be shown by computing the first group cohomology group of $\mathbb{Z}_2$ with coefficients in $\mathbb{Z}_4$, so $H^1(\mathbb{Z}_2,\mathbb{Z}_4) \cong \mathbb{Z}/2$ and noting the two groups in these extensions are $\mathbb{Z}_2\times\mathbb{Z}_4$ and the dihedral group $D_8$. But, as neither of these groups is isomorphic with $Q_8$, the quaternion group is not split. This non-existence of isomorphisms can be checked by noting the trivial extension is abelian while $Q_8$ is non-abelian, and noting the only normal subgroups are $\mathbb{Z}_2$ and $\mathbb{Z}_4$, but $Q_8$ has three subgroups isomorphic to $\mathbb{Z}_4$.
Properties.
If "G" is the semidirect product of the normal subgroup "N" and the subgroup "H", and both "N" and "H" are finite, then the order of "G" equals the product of the orders of "N" and "H". This follows from the fact that "G" is of the same order as the outer semidirect product of "N" and "H", whose underlying set is the Cartesian product "N" × "H".
Relation to direct products.
Suppose "G" is a semidirect product of the normal subgroup "N" and the subgroup "H". If "H" is also normal in "G", or equivalently, if there exists a homomorphism "G" → "N" that is the identity on "N" with kernel "H", then "G" is the direct product of "N" and "H".
The direct product of two groups "N" and "H" can be thought of as the semidirect product of "N" and "H" with respect to "φ"("h")
id"N" for all "h" in "H".
Note that in a direct product, the order of the factors is not important, since "N" × "H" is isomorphic to "H" × "N". This is not the case for semidirect products, as the two factors play different roles.
Furthermore, the result of a (proper) semidirect product by means of a non-trivial homomorphism is never an abelian group, even if the factor groups are abelian.
Non-uniqueness of semidirect products (and further examples).
As opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if "G" and "G′" are two groups that both contain isomorphic copies of "N" as a normal subgroup and "H" as a subgroup, and both are a semidirect product of "N" and "H", then it does "not" follow that "G" and "G′" are isomorphic because the semidirect product also depends on the choice of an action of "H" on "N".
For example, there are four non-isomorphic groups of order 16 that are semidirect products of C8 and C2; in this case, C8 is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups:
If a given group is a semidirect product, then there is no guarantee that this decomposition is unique. For example, there is a group of order 24 (the only one containing six elements of order 4 and six elements of order 6) that can be expressed as semidirect product in the following ways: (D8 ⋉ C3) ≅ (C2 ⋉ Q12) ≅ (C2 ⋉ D12) ≅ (D6 ⋉ V).
Existence.
In general, there is no known characterization (i.e., a necessary and sufficient condition) for the existence of semidirect products in groups. However, some sufficient conditions are known, which guarantee existence in certain cases. For finite groups, the Schur–Zassenhaus theorem guarantees existence of a semidirect product when the order of the normal subgroup is coprime to the order of the quotient group.
For example, the Schur–Zassenhaus theorem implies the existence of a semi-direct product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance.
Generalizations.
Within group theory, the construction of semidirect products can be pushed much further. The Zappa–Szep product of groups is a generalization that, in its internal version, does not assume that either subgroup is normal.
There is also a construction in ring theory, the crossed product of rings. This is constructed in the natural way from the group ring for a semidirect product of groups. The ring-theoretic approach can be further generalized to the semidirect sum of Lie algebras.
For geometry, there is also a crossed product for group actions on a topological space; unfortunately, it is in general non-commutative even if the group is abelian. In this context, the semidirect product is the "space of orbits" of the group action. The latter approach has been championed by Alain Connes as a substitute for approaches by conventional topological techniques; c.f. noncommutative geometry.
The semidirect product is a special case of the Grothendieck construction in category theory. Specifically, an action of $H$ on $N$ (respecting the group, or even just monoid structure) is the same thing as a functor
$F : BH \to Cat$
from the groupoid $BH$ associated to "H" (having a single object *, whose endomorphisms are "H") to the category of categories such that the unique object in $BH$ is mapped to $BN$. The Grothendieck construction of this functor is equivalent to $B(H \rtimes N)$, the (groupoid associated to) semidirect product.
Groupoids.
Another generalization is for groupoids. This occurs in topology because if a group "G" acts on a space "X" it also acts on the fundamental groupoid "π"1("X") of the space. The semidirect product "π"1("X") ⋊ "G" is then relevant to finding the fundamental groupoid of the orbit space "X/G". For full details see Chapter 11 of the book referenced below, and also some details in semidirect product in ncatlab.
Abelian categories.
Non-trivial semidirect products do "not" arise in abelian categories, such as the category of modules. In this case, the splitting lemma shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.
Notation.
Usually the semidirect product of a group "H" acting on a group "N" (in most cases by conjugation as subgroups of a common group) is denoted by "N" ⋊ "H" or "H" ⋉ "N". However, some sources may use this symbol with the opposite meaning. In case the action "φ": "H" → Aut("N") should be made explicit, one also writes "N" ⋊"φ" "H". One way of thinking about the "N" ⋊ "H" symbol is as a combination of the symbol for normal subgroup (◁) and the symbol for the product (×). Barry Simon, in his book on group representation theory, employs the unusual notation $N\mathbin{\circledS_{\varphi}}H$ for the semidirect product.
Unicode lists four variants:
Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice.
In LaTeX, the commands \rtimes and \ltimes produce the corresponding characters. With the AMS symbols package loaded, \leftthreetimes produces ⋋ and \rightthreetimes produces ⋌.
Notes.
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Mathematical connection between field theory and group theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is "solvable by radicals" if its roots may be expressed by a formula involving only integers, nth roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals.
Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting the angle), and characterizing the regular polygons that are constructible (this characterization was previously given by Gauss, but all known proofs that this characterization is complete require Galois theory).
Galois' work was published by Joseph Liouville fourteen years after his death. The theory took longer to become popular among mathematicians and to be well understood.
Galois theory has been generalized to Galois connections and Grothendieck's Galois theory.
Application to classical problems.
The birth and development of Galois theory was caused by the following question, which was one of the main open mathematical questions until the beginning of 19th century:
Does there exist a formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?
The Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot exist. Galois' theory provides a much more complete answer to this question, by explaining why it "is" possible to solve some equations, including all those of degree four or lower, in the above manner, and why it is not possible for most equations of degree five or higher. Furthermore, it provides a means of determining whether a particular equation can be solved that is both conceptually clear and easily expressed as an algorithm.
Galois' theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterization of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of geometry as
History.
Pre-history.
Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, ("x" – "a")("x" – "b") = "x"2 – ("a" + "b")"x" + "ab", where 1, "a" + "b" and "ab" are the elementary polynomials of degree 0, 1 and 2 in two variables.
This was first formalized by the 16th-century French mathematician François Viète, in Viète's formulas, for the case of positive real roots. In the opinion of the 18th-century British mathematician Charles Hutton, the expression of coefficients of a polynomial in terms of the roots (not only for positive roots) was first understood by the 17th-century French mathematician Albert Girard; Hutton writes:
...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.
In this vein, the discriminant is a symmetric function in the roots that reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots. See Discriminant:Nature of the roots for details.
The cubic was first partly solved by the 15–16th-century Italian mathematician Scipione del Ferro, who did not however publish his results; this method, though, only solved one type of cubic equation. This solution was then rediscovered independently in 1535 by Niccolò Fontana Tartaglia, who shared it with Gerolamo Cardano, asking him to not publish it. Cardano then extended this to numerous other cases, using similar arguments; see more details at Cardano's method. After the discovery of del Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he published his solution in his 1545 "Ars Magna." His student Lodovico Ferrari solved the quartic polynomial; his solution was also included in "Ars Magna." In this book, however, Cardano did not provide a "general formula" for the solution of a cubic equation, as he had neither complex numbers at his disposal, nor the algebraic notation to be able to describe a general cubic equation. With the benefit of modern notation and complex numbers, the formulae in this book do work in the general case, but Cardano did not know this. It was Rafael Bombelli who managed to understand how to work with complex numbers in order to solve all forms of cubic equation.
A further step was the 1770 paper "Réflexions sur la résolution algébrique des équations" by the French-Italian mathematician Joseph Louis Lagrange, in his method of Lagrange resolvents, where he analyzed Cardano's and Ferrari's solution of cubics and quartics by considering them in terms of "permutations" of the roots, which yielded an auxiliary polynomial of lower degree, providing a unified understanding of the solutions and laying the groundwork for group theory and Galois' theory. Crucially, however, he did not consider "composition" of permutations. Lagrange's method did not extend to quintic equations or higher, because the resolvent had higher degree.
The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation "groups", not just a single permutation. His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of the Norwegian mathematician Niels Henrik Abel, who published a proof in 1824, thus establishing the Abel–Ruffini theorem.
While Ruffini and Abel established that the "general" quintic could not be solved, some "particular" quintics can be solved, such as "x"5 - 1
0, and the precise criterion by which a "given" quintic or higher polynomial could be determined to be solvable or not was given by Évariste Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its Galois group – had a certain structure – in modern terms, whether or not it was a solvable group. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degrees.
Galois' writings.
In 1830 Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals; Galois' paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois then died in a duel in 1832, and his paper, "Mémoire sur les conditions de résolubilité des équations par radicaux", remained unpublished until 1846 when it was published by Joseph Liouville accompanied by some of his own explanations. Prior to this publication, Liouville announced Galois' result to the Academy in a speech he gave on 4 July 1843. According to Allan Clark, Galois's characterization "dramatically supersedes the work of Abel and Ruffini."
Aftermath.
Galois' theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it. For example, in his 1846 commentary, Liouville completely missed the group-theoretic core of Galois' method. Joseph Alfred Serret who attended some of Liouville's talks, included Galois' theory in his 1866 (third edition) of his textbook "Cours d'algèbre supérieure". Serret's pupil, Camille Jordan, had an even better understanding reflected in his 1870 book "Traité des substitutions et des équations algébriques". Outside France, Galois' theory remained more obscure for a longer period. In Britain, Cayley failed to grasp its depth and popular British algebra textbooks did not even mention Galois' theory until well after the turn of the century. In Germany, Kronecker's writings focused more on Abel's result. Dedekind wrote little about Galois' theory, but lectured on it at Göttingen in 1858, showing a very good understanding. Eugen Netto's books of the 1880s, based on Jordan's "Traité", made Galois theory accessible to a wider German and American audience as did Heinrich Martin Weber's 1895 algebra textbook.
Permutation group approach.
Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say "A" and "B", that "A"2 + 5"B"3 = 7. The central idea of Galois' theory is to consider permutations (or rearrangements) of the roots such that "any" algebraic equation satisfied by the roots is "still satisfied" after the roots have been permuted. Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers. It extends naturally to equations with coefficients in any field, but this will not be considered in the simple examples below.
These permutations together form a permutation group, also called the Galois group of the polynomial, which is explicitly described in the following examples.
Quadratic equation.
Consider the quadratic equation
$x^2 - 4x + 1 = 0. $
By using the quadratic formula, we find that the two roots are
$\begin{align}
A &= 2 + \sqrt{3},\\
B &= 2 - \sqrt{3}.
\end{align}$
Examples of algebraic equations satisfied by "A" and "B" include
$A + B = 4, $
and
$AB = 1. $
If we exchange "A" and "B" in either of the last two equations we obtain another true statement. For example, the equation "A" + "B"
4 becomes "B" + "A"
4. It is more generally true that this holds for "every" possible algebraic relation between "A" and "B" such that all coefficients are rational; that is, in any such relation, swapping "A" and "B" yields another true relation. This results from the theory of symmetric polynomials, which, in this case, may be replaced by formula manipulations involving the binomial theorem.
One might object that "A" and "B" are related by the algebraic equation "A" − "B" − 2√3
0, which does not remain true when "A" and "B" are exchanged. However, this relation is not considered here, because it has the coefficient which is not rational.
We conclude that the Galois group of the polynomial "x"2 − 4"x" + 1 consists of two permutations: the identity permutation which leaves "A" and "B" untouched, and the transposition permutation which exchanges "A" and "B". As all groups with two elements are isomorphic, this Galois group is isomorphic to the multiplicative group {1, −1}.
A similar discussion applies to any quadratic polynomial "ax"2 + "bx" + "c", where "a", "b" and "c" are rational numbers.
("x" − 2)2, or "x"2 − 3"x" + 2
("x" − 2)("x" − 1), then the Galois group is trivial; that is, it contains only the identity permutation. In this example, if "A"
2 and "B"
1 then "A" − "B"
1 is no longer true when "A" and "B" are swapped.
Quartic equation.
Consider the polynomial
$x^4 - 10x^2 + 1,$
which can also be written as
$\left(x^2 - 5\right)^2 - 24.$
We wish to describe the Galois group of this polynomial, again over the field of rational numbers. The polynomial has four roots:
$\begin{align}
A &= \sqrt{2} + \sqrt{3},\\
B &= \sqrt{2} - \sqrt{3},\\
C &= -\sqrt{2} + \sqrt{3},\\
D &= -\sqrt{2} - \sqrt{3}.
\end{align}$
There are 24 possible ways to permute these four roots, but not all of these permutations are members of the Galois group. The members of the Galois group must preserve any algebraic equation with rational coefficients involving "A", "B", "C" and "D".
Among these equations, we have:
$\begin{align}
AB&=-1 \\ AC&=1 \\ A+D&=0
\end{align}$
It follows that, if "φ" is a permutation that belongs to the Galois group, we must have:
$\begin{align}
\varphi(B)&=\frac{-1}{\varphi(A)}, \\ \varphi(C)&=\frac{1}{\varphi(A)}, \\ \varphi(D)&=-\varphi(A).
\end{align}$
This implies that the permutation is well defined by the image of "A", and that the Galois group has 4 elements, which are:
("A", "B", "C", "D") → ("A", "B", "C", "D")
("A", "B", "C", "D") → ("B", "A", "D", "C")
("A", "B", "C", "D") → ("C", "D", "A", "B")
("A", "B", "C", "D") → ("D", "C", "B", "A")
This implies that the Galois group is isomorphic to the Klein four-group.
Modern approach by field theory.
In the modern approach, one starts with a field extension "L"/"K" (read ""L" over "K""), and examines the group of automorphisms of "L" that fix "K". See the article on Galois groups for further explanation and examples.
The connection between the two approaches is as follows. The coefficients of the polynomial in question should be chosen from the base field "K". The top field "L" should be the field obtained by adjoining the roots of the polynomial in question to the base field. Any permutation of the roots which respects algebraic equations as described above gives rise to an automorphism of "L"/"K", and vice versa.
In the first example above, we were studying the extension Q(√3)/Q, where Q is the field of rational numbers, and Q(√3) is the field obtained from Q by adjoining . In the second example, we were studying the extension Q("A","B","C","D")/Q.
There are several advantages to the modern approach over the permutation group approach.
Solvable groups and solution by radicals.
The notion of a solvable group in group theory allows one to determine whether a polynomial is solvable in radicals, depending on whether its Galois group has the property of solvability. In essence, each field extension "L"/"K" corresponds to a factor group in a composition series of the Galois group. If a factor group in the composition series is cyclic of order "n", and if in the corresponding field extension "L"/"K" the field "K" already contains a primitive "n"th root of unity, then it is a radical extension and the elements of "L" can then be expressed using the "n"th root of some element of "K".
If all the factor groups in its composition series are cyclic, the Galois group is called "solvable", and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field (usually Q).
One of the great triumphs of Galois Theory was the proof that for every "n" > 4, there exist polynomials of degree "n" which are not solvable by radicals (this was proven independently, using a similar method, by Niels Henrik Abel a few years before, and is the Abel–Ruffini theorem), and a systematic way for testing whether a specific polynomial is solvable by radicals. The Abel–Ruffini theorem results from the fact that for "n" > 4 the symmetric group "S""n" contains a simple, noncyclic, normal subgroup, namely the alternating group "A""n".
A non-solvable quintic example.
Van der Waerden cites the polynomial "f"("x")
"x"5 − "x" − 1. By the rational root theorem this has no rational zeroes. Neither does it have linear factors modulo 2 or 3.
The Galois group of "f"("x") modulo 2 is cyclic of order 6, because "f"("x") modulo 2 factors into polynomials of orders 2 and 3, ("x"2 + "x" + 1)("x"3 + "x"2 + 1).
"f"("x") modulo 3 has no linear or quadratic factor, and hence is irreducible. Thus its modulo 3 Galois group contains an element of order 5.
It is known that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. A permutation group on 5 objects with elements of orders 6 and 5 must be the symmetric group "S"5, which is therefore the Galois group of "f"("x"). This is one of the simplest examples of a non-solvable quintic polynomial. According to Serge Lang, Emil Artin was fond of this example.
Inverse Galois problem.
The "inverse Galois problem" is to find a field extension with a given Galois group.
As long as one does not also specify the ground field, the problem is not very difficult, and all finite groups do occur as Galois groups.
For showing this, one may proceed as follows. Choose a field "K" and a finite group "G". Cayley's theorem says that "G" is (up to isomorphism) a subgroup of the symmetric group "S" on the elements of "G". Choose indeterminates {"x""α"}, one for each element "α" of "G", and adjoin them to "K" to get the field "F"
"K"({"x""α"}). Contained within "F" is the field "L" of symmetric rational functions in the {"x""α"}. The Galois group of "F"/"L" is "S", by a basic result of Emil Artin. "G" acts on "F" by restriction of action of "S". If the fixed field of this action is "M", then, by the fundamental theorem of Galois theory, the Galois group of "F"/"M" is "G".
On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field Q of the rational numbers. Igor Shafarevich proved that every solvable finite group is the Galois group of some extension of Q. Various people have solved the inverse Galois problem for selected non-Abelian simple groups. Existence of solutions has been shown for all but possibly one (Mathieu group "M"23) of the 26 sporadic simple groups. There is even a polynomial with integral coefficients whose Galois group is the Monster group.
Inseparable extensions.
In the form mentioned above, including in particular the fundamental theorem of Galois theory, the theory only considers Galois extensions, which are in particular separable. General field extensions can be split into a separable, followed by a purely inseparable field extension. For a purely inseparable extension "F" / "K", there is a Galois theory where the Galois group is replaced by the vector space of derivations, $Der_K(F, F)$, i.e., "K"-linear endomorphisms of "F" satisfying the Leibniz rule. In this correspondence, an intermediate field "E" is assigned $Der_E(F, F) \subset Der_K(F, F)$. Conversely, a subspace $V \subset Der_K(F, F)$ satisfying appropriate further conditions is mapped to $\{x \in F, f(x)=0\ \forall f \in V\}$. Under the assumption $F^p \subset K$, showed that this establishes a one-to-one correspondence. The condition imposed by Jacobson has been removed by , by giving a correspondence using notions of derived algebraic geometry.
Notes.
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Alternative mathematical ordering
In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as ""a" < "b"". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation ["a", "b", "c"], meaning "after a, one reaches b before c". For example, [June, October, February], but not [June, February, October], cf. picture. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected. Dropping the "connected" requirement results in a partial cyclic order.
A set with a cyclic order is called a cyclically ordered set or simply a cycle.[nb] Some familiar cycles are discrete, having only a finite number of elements: there are seven days of the week, four cardinal directions, twelve notes in the chromatic scale, and three plays in rock-paper-scissors. In a finite cycle, each element has a "next element" and a "previous element". There are also cyclic orders with infinitely many elements, such as the oriented unit circle in the plane.
Cyclic orders are closely related to the more familiar linear orders, which arrange objects in a line. Any linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line. These operations, along with the related constructions of intervals and covering maps, mean that questions about cyclic orders can often be transformed into questions about linear orders. Cycles have more symmetries than linear orders, and they often naturally occur as residues of linear structures, as in the finite cyclic groups or the real projective line.
Finite cycles.
A cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock. Each element x in X has a "next element" and a "previous element", and taking either successors or predecessors cycles exactly once through the elements as "x"(1), "x"(2), ..., "x"("n").
There are a few equivalent ways to state this definition. A cyclic order on X is the same as a permutation that makes all of X into a single cycle. A cycle with n elements is also a Z"n"-torsor: a set with a free transitive action by a finite cyclic group. Another formulation is to make X into the standard directed cycle graph on n vertices, by some matching of elements to vertices.
It can be instinctive to use cyclic orders for symmetric functions, for example as in
"xy" + "yz" + "zx"
where writing the final monomial as "xz" would distract from the pattern.
A substantial use of cyclic orders is in the determination of the conjugacy classes of free groups. Two elements g and h of the free group F on a set Y are conjugate if and only if, when they are written as products of elements y and "y−1" with y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the rewriting rules that allow one to remove or add adjacent y and "y"−1.
A cyclic order on a set X can be determined by a linear order on X, but not in a unique way. Choosing a linear order is equivalent to choosing a first element, so there are exactly n linear orders that induce a given cyclic order. Since there are "n"! possible linear orders, there are ("n" − 1)! possible cyclic orders.
Definitions.
An infinite set can also be ordered cyclically. Important examples of infinite cycles include the unit circle, "S"1, and the rational numbers, Q. The basic idea is the same: we arrange elements of the set around a circle. However, in the infinite case we cannot rely upon an immediate successor relation, because points may not have successors. For example, given a point on the unit circle, there is no "next point". Nor can we rely upon a binary relation to determine which of two points comes "first". Traveling clockwise on a circle, neither east or west comes first, but each follows the other.
Instead, we use a ternary relation denoting that elements a, b, c occur after each other (not necessarily immediately) as we go around the circle. For example, in clockwise order, [east, south, west]. By currying the arguments of the ternary relation ["a", "b", "c"], one can think of a cyclic order as a one-parameter family of binary order relations, called "cuts", or as a two-parameter family of subsets of K, called "intervals".
The ternary relation.
The general definition is as follows: a cyclic order on a set X is a relation "C" ⊂ "X"3, written ["a", "b", "c"], that satisfies the following axioms:[nb]
The axioms are named by analogy with the asymmetry, transitivity, and connectedness axioms for a binary relation, which together define a strict linear order. Edward Huntington (1916, 1924) considered other possible lists of axioms, including one list that was meant to emphasize the similarity between a cyclic order and a betweenness relation. A ternary relation that satisfies the first three axioms, but not necessarily the axiom of totality, is a partial cyclic order.
Rolling and cuts.
Given a linear order < on a set X, the cyclic order on X induced by < is defined as follows:
["a", "b", "c"] if and only if "a" < "b" < "c" or "b" < "c" < "a" or "c" < "a" < "b"
Two linear orders induce the same cyclic order if they can be transformed into each other by a cyclic rearrangement, as in
cutting a deck of cards. One may define a cyclic order relation as a ternary relation that is induced by a strict linear order as above.
Cutting a single point out of a cyclic order leaves a linear order behind. More precisely, given a cyclically ordered set ("K", [ ]), each element "a" ∈ "K" defines a natural linear order <"a" on the remainder of the set, "K" ∖ "a", by the following rule:
"x" <"a" "y" if and only if ["a", "x", "y"].
Moreover, <"a" can be extended by adjoining a as a least element; the resulting linear order on K is called the principal cut with least element a. Likewise, adjoining a as a greatest element results in a cut <"a".
Intervals.
Given two elements "a" ≠ "b" ∈ "K", the open interval from a to b, written ("a", "b"), is the set of all "x" ∈ "K" such that ["a", "x", "b"]. The system of open intervals completely defines the cyclic order and can be used as an alternate definition of a cyclic order relation.
An interval ("a", "b") has a natural linear order given by <"a". One can define half-closed and closed intervals ["a", "b"), ("a", "b"], and ["a", "b"] by adjoining a as a least element and/or b as a greatest element. As a special case, the open interval ("a", "a") is defined as the cut "K" ∖ "a".
More generally, a proper subset "S" of "K" is called convex if it contains an interval between every pair of points: for "a" ≠ "b" ∈ "S", either ("a", "b") or ("b", "a") must also be in "S". A convex set is linearly ordered by the cut <"x" for any x not in the set; this ordering is independent of the choice of x.
Automorphisms.
As a circle has a clockwise order and a counterclockwise order, any set with a cyclic order has two senses. A bijection of the set that preserves the order is called an ordered correspondence. If the sense is maintained as before, it is a direct correspondence, otherwise it is called an opposite correspondence. Coxeter uses a separation relation to describe cyclic order, and this relation is strong enough to distinguish the two senses of cyclic order. The automorphisms of a cyclically ordered set may be identified with C2, the two-element group, of direct and opposite correspondences.
Monotone functions.
The "cyclic order = arranging in a circle" idea works because any subset of a cycle is itself a cycle. In order to use this idea to impose cyclic orders on sets that are not actually subsets of the unit circle in the plane, it is necessary to consider functions between sets.
A function between two cyclically ordered sets, "f" : "X" → "Y", is called a "monotonic function" or a "homomorphism" if it pulls back the ordering on Y: whenever ["f"("a"), "f"("b"), "f"("c")], one has ["a", "b", "c"]. Equivalently, f is monotone if whenever ["a", "b", "c"] and "f"("a"), "f"("b"), and "f"("c") are all distinct, then ["f"("a"), "f"("b"), "f"("c")]. A typical example of a monotone function is the following function on the cycle with 6 elements:
"f"(0) = "f"(1) = 4,
"f"(2) = "f"(3) = 0,
"f"(4) = "f"(5) = 1.
A function is called an "embedding" if it is both monotone and injective.[nb] Equivalently, an embedding is a function that pushes forward the ordering on X: whenever ["a", "b", "c"], one has ["f"("a"), "f"("b"), "f"("c")]. As an important example, if X is a subset of a cyclically ordered set Y, and X is given its natural ordering, then the inclusion map "i" : "X" → "Y" is an embedding.
Generally, an injective function f from an unordered set X to a cycle Y induces a unique cyclic order on X that makes f an embedding.
Functions on finite sets.
A cyclic order on a finite set X can be determined by an injection into the unit circle, "X" → "S"1. There are many possible functions that induce the same cyclic order—in fact, infinitely many. In order to quantify this redundancy, it takes a more complex combinatorial object than a simple number. Examining the configuration space of all such maps leads to the definition of an polytope known as a cyclohedron. Cyclohedra were first applied to the study of knot invariants; they have more recently been applied to the experimental detection of periodically expressed genes in the study of biological clocks.
The category of homomorphisms of the standard finite cycles is called the cyclic category; it may be used to construct Alain Connes' cyclic homology.
One may define a degree of a function between cycles, analogous to the degree of a continuous mapping. For example, the natural map from the circle of fifths to the chromatic circle is a map of degree 7. One may also define a rotation number.
Completion.
The set of all cuts is cyclically ordered by the following relation: [<1, <2, <3] if and only if there exist "x", "y", "z" such that:
"x" <1 "y" <1 "z",
"x" <1"y" <2 "z" <2 "x", and
"x" <1 "y" <1"z" <3 "x" <3 "y".
A certain subset of this cycle of cuts is the Dedekind completion of the original cycle.
Further constructions.
Unrolling and covers.
Starting from a cyclically ordered set K, one may form a linear order by unrolling it along an infinite line. This captures the intuitive notion of keeping track of how many times one goes around the circle. Formally, one defines a linear order on the Cartesian product Z × "K", where Z is the set of integers, by fixing an element a and requiring that for all i:
If ["a", "x", "y"], then "a""i" < "x""i" < "y""i" < "a""i" + 1.
For example, the months 2023, 2023, 2023, and 2024 occur in that order.
This ordering of Z × "K" is called the universal cover of K.[nb] Its order type is independent of the choice of a, but the notation is not, since the integer coordinate "rolls over" at a. For example, although the cyclic order of pitch classes is compatible with the A-to-G alphabetical order, C is chosen to be the first note in each octave, so in note-octave notation, B3 is followed by C4.
The inverse construction starts with a linearly ordered set and coils it up into a cyclically ordered set. Given a linearly ordered set L and an order-preserving bijection "T" : "L" → "L" with unbounded orbits, the orbit space "L" / "T" is cyclically ordered by the requirement:[nb]
If "a" < "b" < "c" < "T"("a"), then "a"], ["b"], ["c".
In particular, one can recover K by defining "T"("x""i") = "x""i" + 1 on Z × "K".
There are also n-fold coverings for finite n; in this case, one cyclically ordered set covers another cyclically ordered set. For example, the 24-hour clock is a double cover of the 12-hour clock. In geometry, the pencil of rays emanating from a point in the oriented plane is a double cover of the pencil of unoriented lines passing through the same point. These covering maps can be characterized by lifting them to the universal cover.
Products and retracts.
Given a cyclically ordered set ("K", [ ]) and a linearly ordered set ("L", <), the (total) lexicographic product is a cyclic order on the product set "K" × "L", defined by [("a", "x"), ("b", "y"), ("c", "z")] if one of the following holds:
The lexicographic product "K" × "L" globally looks like K and locally looks like L; it can be thought of as K copies of L. This construction is sometimes used to characterize cyclically ordered groups.
One can also glue together different linearly ordered sets to form a circularly ordered set. For example, given two linearly ordered sets "L"1 and "L"2, one may form a circle by joining them together at positive and negative infinity. A circular order on the disjoint union "L"1 ∪ "L"2 ∪ {–∞, ∞} is defined by ∞ < "L"1 < –∞ < "L"2 < ∞, where the induced ordering on "L"1 is the opposite of its original ordering. For example, the set of all longitudes is circularly ordered by joining all points west and all points east, along with the prime meridian and the 180th meridian. use this construction while characterizing the spaces of orderings and real places of double formal Laurent series over a real closed field.
Topology.
The open intervals form a base for a natural topology, the cyclic order topology. The open sets in this topology are exactly those sets which are open in "every" compatible linear order. To illustrate the difference, in the set [0, 1), the subset [0, 1/2) is a neighborhood of 0 in the linear order but not in the cyclic order.
Interesting examples of cyclically ordered spaces include the conformal boundary of a simply connected Lorentz surface and the leaf space of a lifted essential lamination of certain 3-manifolds. Discrete dynamical systems on cyclically ordered spaces have also been studied.
The interval topology forgets the original orientation of the cyclic order. This orientation can be restored by enriching the intervals with their induced linear orders; then one has a set covered with an atlas of linear orders that are compatible where they overlap. In other words, a cyclically ordered set can be thought of as a locally linearly ordered space: an object like a manifold, but with order relations instead of coordinate charts. This viewpoint makes it easier to be precise about such concepts as covering maps. The generalization to a locally partially ordered space is studied in ; see also "Directed topology".
Related structures.
Groups.
A cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947. They are a generalization of cyclic groups: the infinite cyclic group Z and the finite cyclic groups Z/"n". Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers Q, the real numbers R, and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group T and its subgroups, such as the subgroup of rational points.
Every cyclically ordered group can be expressed as a quotient "L" / "Z", where L is a linearly ordered group and Z is a cyclic cofinal subgroup of L. Every cyclically ordered group can also be expressed as a subgroup of a product T × "L", where L is a linearly ordered group. If a cyclically ordered group is Archimedean or compact, it can be embedded in T itself.
Modified axioms.
A partial cyclic order is a ternary relation that generalizes a (total) cyclic order in the same way that a partial order generalizes a total order. It is cyclic, asymmetric, and transitive, but it need not be total. An order variety is a partial cyclic order that satisfies an additional "spreading" axiom . Replacing the asymmetry axiom with a complementary version results in the definition of a "co-cyclic order". Appropriately total co-cyclic orders are related to cyclic orders in the same way that ≤ is related to <.
A cyclic order obeys a relatively strong 4-point transitivity axiom. One structure that weakens this axiom is a CC system: a ternary relation that is cyclic, asymmetric, and total, but generally not transitive. Instead, a CC system must obey a 5-point transitivity axiom and a new "interiority" axiom, which constrains the 4-point configurations that violate cyclic transitivity.
A cyclic order is required to be symmetric under cyclic permutation, ["a", "b", "c"] ⇒ ["b", "c", "a"], and asymmetric under reversal: ["a", "b", "c"] ⇒ ¬["c", "b", "a"]. A ternary relation that is "asymmetric" under cyclic permutation and "symmetric" under reversal, together with appropriate versions of the transitivity and totality axioms, is called a betweenness relation. A separation relation is a quaternary relation that can be thought of as a cyclic order without an orientation. The relationship between a circular order and a separation relation is analogous to the relationship between a linear order and a betweenness relation.
Symmetries and model theory.
provide a model-theoretic description of the covering maps of cycles.
Tararin (2001, 2002) studies groups of automorphisms of cycles with various transitivity properties. characterize cycles whose full automorphism groups act freely and transitively. characterize countable colored cycles whose automorphism groups act transitively. studies the automorphism group of the unique (up to isomorphism) countable dense cycle.
study minimality conditions on circularly ordered structures, i.e. models of first-order languages that include a cyclic order relation. These conditions are analogues of o-minimality and weak o-minimality for the case of linearly ordered structures. Kulpeshov (2006, 2009) continues with some characterizations of ω-categorical structures.
Cognition.
Hans Freudenthal has emphasized the role of cyclic orders in cognitive development, as a contrast to Jean Piaget who addresses only linear orders. Some experiments have been performed to investigate the mental representations of cyclically ordered sets, such as the months of the year.
Notes on usage.
^cyclic order The relation may be called a "cyclic order" , a "circular order" , a "cyclic ordering" , or a "circular ordering" . Some authors call such an ordering a "total cyclic order" , a "complete cyclic order" , a "linear cyclic order" , or an "l-cyclic order" or ℓ-"cyclic order" , to distinguish from the broader class of partial cyclic orders, which they call simply "cyclic orders". Finally, some authors may take "cyclic order" to mean an unoriented quaternary separation relation .
^cycle A set with a cyclic order may be called a "cycle" or a "circle" . The above variations also appear in adjective form: "cyclically ordered set" ("cyklicky uspořádané množiny", ), "circularly ordered set", "total cyclically ordered set", "complete cyclically ordered set", "linearly cyclically ordered set", "l-cyclically ordered set", ℓ-"cyclically ordered set". All authors agree that a cycle is totally ordered.
^ternary relation There are a few different symbols in use for a cyclic relation. uses concatenation: "ABC". and use ordered triples and the set membership symbol: ("a", "b", "c") ∈ "C". uses concatenation and set membership: "abc" ∈ "C", understanding "abc" as a cyclically ordered triple. The literature on groups, such as and , tend to use square brackets: ["a", "b", "c"]. use round parentheses: ("a", "b", "c"), reserving square brackets for a betweenness relation. use a function-style notation: "R"("a", "b", "c"). Rieger (1947), cited after ) uses a "less-than" symbol as a delimiter: < "x", "y", "z" <. Some authors use infix notation: "a" < "b" < "c", with the understanding that this does not carry the usual meaning of "a" < "b" and "b" < "c" for some binary relation < . emphasizes the cyclic nature by repeating an element: "p" ↪ "r" ↪ "q" ↪ "p".
^embedding calls an embedding an "isomorphic embedding".
^roll In this case, write that K is L "rolled up".
^orbit space The map "T" is called "archimedean" by , "coterminal" by , and a "translation" by .
^universal cover calls Z × "K" the "universal cover" of K. write that K is Z × "K" "coiled". call Z × "K" the "∞-times covering" of K. Often this construction is written as the anti-lexicographic order on "K" × Z.
|
180909
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abstract_algebra
|
In mathematics, and in particular in combinatorics, the combinatorial number system of degree "k" (for some positive integer "k"), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) "N" and "k"-combinations. The combinations are represented as strictly decreasing sequences "c""k" > ... > "c"2 > "c"1 ≥ 0 where each "ci" corresponds to the index of a chosen element in a given "k"-combination. Distinct numbers correspond to distinct "k"-combinations, and produce them in lexicographic order. The numbers less than $\tbinom nk$ correspond to all "k"-combinations of {0, 1, ..., "n" − 1}. The correspondence does not depend on the size "n" of the set that the "k"-combinations are taken from, so it can be interpreted as a map from N to the "k"-combinations taken from N; in this view the correspondence is a bijection.
The number "N" corresponding to ("c""k", ..., "c"2, "c"1) is given by
$N=\binom{c_k}k+\cdots+\binom{c_2}2+\binom{c_1}1$.
The fact that a unique sequence corresponds to any non-negative number "N" was first observed by D. H. Lehmer. Indeed, a greedy algorithm finds the "k"-combination corresponding to "N": take "c""k" maximal with $\tbinom{c_k}k\leq N$, then take "c""k"−1 maximal with $\tbinom{c_{k-1}}{k-1}\leq N - \tbinom{c_k}k$, and so forth. Finding the number "N", using the formula above, from the "k"-combination ("c""k", ..., "c"2, "c"1) is also known as "ranking", and the opposite operation (given by the greedy algorithm) as "unranking"; the operations are known by these names in most computer algebra systems, and in computational mathematics.
The originally used term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth,
who also gives a much older reference;
the term "combinadic" is introduced by James McCaffrey (without reference to previous terminology or work).
Unlike the factorial number system, the combinatorial number system of degree "k" is not a mixed radix system: the part $\tbinom{c_i}i$ of the number "N" represented by a "digit" "c""i" is not obtained from it by simply multiplying by a place value.
The main application of the combinatorial number system is that it allows rapid computation of the "k"-combination that is at a given position in the lexicographic ordering, without having to explicitly list the "k"-combinations preceding it; this allows for instance random generation of "k"-combinations of a given set. Enumeration of "k"-combinations has many applications, among which are software testing, sampling, quality control, and the analysis of lottery games.
Ordering combinations.
A "k"-combination of a set "S" is a subset of "S" with "k" (distinct) elements. The main purpose of the combinatorial number system is to provide a representation, each by a single number, of all $\tbinom nk$ possible "k"-combinations of a set "S" of "n" elements. Choosing, for any "n", {0, 1, ..., "n" − 1} as such a set, it can be arranged that the representation of a given "k"-combination "C" is independent of the value of "n" (although "n" must of course be sufficiently large); in other words considering "C" as a subset of a larger set by increasing "n" will not change the number that represents "C". Thus for the combinatorial number system one just considers "C" as a "k"-combination of the set N of all natural numbers, without explicitly mentioning "n".
In order to ensure that the numbers representing the "k"-combinations of {0, 1, ..., "n" − 1} are less than those representing "k"-combinations not contained in {0, 1, ..., "n" − 1}, the "k"-combinations must be ordered in such a way that their largest elements are compared first. The most natural ordering that has this property is lexicographic ordering of the "decreasing" sequence of their elements. So comparing the 5-combinations "C" = {0,3,4,6,9} and "C"′ = {0,1,3,7,9}, one has that "C" comes before "C"′, since they have the same largest part 9, but the next largest part 6 of "C" is less than the next largest part 7 of "C"′; the sequences compared lexicographically are (9,6,4,3,0) and (9,7,3,1,0).
Another way to describe this ordering is view combinations as describing the "k" raised bits in the binary representation of a number, so that "C" = {"c"1, ..., "c""k"} describes the number
$2^{c_1}+2^{c_2}+\cdots+2^{c_k}$
(this associates distinct numbers to "all" finite sets of natural numbers); then comparison of "k"-combinations can be done by comparing the associated binary numbers. In the example "C" and "C"′ correspond to numbers 10010110012 = 60110 and 10100010112 = 65110, which again shows that "C" comes before "C"′. This number is not however the one one wants to represent the "k"-combination with, since many binary numbers have a number of raised bits different from "k"; one wants to find the relative position of "C" in the ordered list of (only) "k"-combinations.
Place of a combination in the ordering.
The number associated in the combinatorial number system of degree "k" to a "k"-combination "C" is the number of "k"-combinations strictly less than "C" in the given ordering. This number can be computed from "C" = {"c""k", ..., "c"2, "c"1} with "c""k" > ... > "c"2 > "c"1 as follows.
From the definition of the ordering it follows that for each "k"-combination "S" strictly less than "C", there is a unique index "i" such that "c""i" is absent from "S", while "c""k", ..., "c""i"+1 are present in "S", and no other value larger than "c""i" is. One can therefore group those "k"-combinations "S" according to the possible values 1, 2, ..., "k" of "i", and count each group separately. For a given value of "i" one must include
"c""k", ..., "c""i"+1 in "S", and the remaining "i" elements of "S" must be chosen from the "c""i" non-negative integers strictly less than "c""i"; moreover any such choice will result in a "k"-combinations "S" strictly less than "C". The number of possible choices is $\tbinom{c_i}i$, which is therefore the number of combinations in group "i"; the total number of "k"-combinations strictly less than "C" then is
$\binom{c_1}1+\binom{c_2}2+\cdots+\binom{c_k}k,$
and this is the index (starting from 0) of "C" in the ordered list of "k"-combinations.
Obviously there is for every "N" ∈ N exactly one "k"-combination at index "N" in the list (supposing "k" ≥ 1, since the list is then infinite), so the above argument proves that every "N" can be written in exactly one way as a sum of "k" binomial coefficients of the given form.
Finding the "k"-combination for a given number.
The given formula allows finding the place in the lexicographic ordering of a given "k"-combination immediately. The reverse process of finding the "k"-combination at a given place "N" requires somewhat more work, but is straightforward nonetheless. By the definition of the lexicographic ordering, two "k"-combinations that differ in their largest element "c""k" will be ordered according to the comparison of those largest elements, from which it follows that all combinations with a fixed value of their largest element are contiguous in the list. Moreover the smallest combination with "c""k" as the largest element is $\tbinom{c_k}k$, and it has "c""i" = "i" − 1 for all "i" < "k" (for this combination all terms in the expression except $\tbinom{c_k}k$ are zero). Therefore "c""k" is the largest number such that $\tbinom{c_k}k\leq N$. If "k" > 1 the remaining elements of the "k"-combination form the "k" − 1-combination corresponding to the number $N-\tbinom{c_k}k$ in the combinatorial number system of degree "k" − 1, and can therefore be found by continuing in the same way for $N-\tbinom{c_k}k$ and "k" − 1 instead of "N" and "k".
Example.
Suppose one wants to determine the 5-combination at position 72. The successive values of $\tbinom n5$ for "n" = 4, 5, 6, ... are 0, 1, 6, 21, 56, 126, 252, ..., of which the largest one not exceeding 72 is 56, for "n" = 8. Therefore "c"5 = 8, and the remaining elements form the 4-combination at position 72 − 56
16. The successive values of $\tbinom n4$ for "n" = 3, 4, 5, ... are 0, 1, 5, 15, 35, ..., of which the largest one not exceeding 16 is 15, for "n" = 6, so "c"4 = 6. Continuing similarly to search for a 3-combination at position 16 − 15
1 one finds "c"3 = 3, which uses up the final unit; this establishes $72=\tbinom85+\tbinom64+\tbinom33$, and the remaining values "c""i" will be the maximal ones with $\tbinom{c_i}i=0$, namely "c""i"
"i" − 1. Thus we have found the 5-combination {8, 6, 3, 1, 0}.
National Lottery example.
For each of the $\binom{49}6$ lottery combinations "c"1 < "c"2 < "c"3 < "c"4 < "c"5 < "c"6 , there is a list number "N" between 0 and $\binom{49}6 - 1$ which can be found by adding
$ \binom{49-c_1} 6 + \binom{49-c_2} 5 + \binom{49-c_3} 4 + \binom{49-c_4} 3 + \binom{49-c_5} 2 + \binom{49-c_6} 1. $
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348399
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abstract_algebra
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Method of graph decomposition
In graph theory, a bramble for an undirected graph G is a family of connected subgraphs of G that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in G that has one endpoint in each subgraph. The "order" of a bramble is the smallest size of a hitting set, a set of vertices of G that has a nonempty intersection with each of the subgraphs. Brambles may be used to characterize the treewidth of G.
Treewidth and havens.
A haven of order "k" in a graph "G" is a function "β" that maps each set "X" of fewer than "k" vertices to a connected component of "G" − "X", in such a way that every two subsets "β"("X") and "β"("Y") touch each other. Thus, the set of images of "β" forms a bramble in "G", with order "k". Conversely, every bramble may be used to determine a haven: for each set "X" of size smaller than the order of the bramble, there is a unique connected component "β"("X") that contains all of the subgraphs in the bramble that are disjoint from "X".
As Seymour and Thomas showed, the order of a bramble (or equivalently, of a haven) characterizes treewidth: a graph has a bramble of order "k" if and only if it has treewidth at least "k" − 1.
Size of brambles.
Expander graphs of bounded degree have treewidth proportional to their number of vertices, and therefore also have brambles of linear order. However, as Martin Grohe and Dániel Marx showed, for these graphs, a bramble of such a high order must include exponentially many sets. More strongly, for these graphs, even brambles whose order is slightly larger than the square root of the treewidth must have exponential size. However, Grohe and Marx also showed that every graph of treewidth "k" has a bramble of polynomial size and of order $\Omega(k^{1/2}/\log^2 k)$.
Computational complexity.
Because brambles may have exponential size, it is not always possible to construct them in polynomial time for graphs of unbounded treewidth. However, when the treewidth is bounded, a polynomial time construction is possible: it is possible to find a bramble of order "k", when one exists, in time O("n""k" + 2) where "n" is the number of vertices in the given graph. Even faster algorithms are possible for graphs with few minimal separators.
Bodlaender, Grigoriev, and Koster studied heuristics for finding brambles of high order. Their methods do not always generate brambles of order close to the treewidth of the input graph, but for planar graphs they give a constant approximation ratio. Kreutzer and Tazari provide randomized algorithms that, on graphs of treewidth "k", find brambles of polynomial size and of order $\Omega(k^{1/2}/\log^3 k)$ within polynomial time, coming within a logarithmic factor of the order shown by to exist for polynomial size brambles.
Directed brambles.
The concept of bramble has also been defined for directed graphs. In a directed graph "D", a bramble is a collection of strongly connected subgraphs of "D" that all touch each other: for every pair of disjoint elements "X", "Y" of the bramble, there must exist an arc in "D" from "X" to "Y" and one from "Y" to "X". The "order" of a bramble is the smallest size of a hitting set, a set of vertices of "D" that has a nonempty intersection with each of the elements of the bramble. The "bramble number" of "D" is the maximum order of a bramble in "D".
The bramble number of a directed graph is within a constant factor of its directed treewidth.
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3829452
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abstract_algebra
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Non-abelian group of order eight
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
$\{1,i,j,k,-1,-i,-j,-k\}$ of the quaternions under multiplication. It is given by the group presentation
$\mathrm{Q}_8 = \langle \bar{e},i,j,k \mid \bar{e}^2 = e, \;i^2 = j^2 = k^2 = ijk = \bar{e} \rangle ,$
where "e" is the identity element and "e" commutes with the other elements of the group.
Another presentation of Q8 is
$\mathrm{Q}_8 = \langle a,b \mid a^4 = e, a^2 = b^2, ba = a^{-1}b\rangle.$
The quaternion group sprang full-blown from the mind of W. R. Hamilton, and there has been an effort to connect it with the wellspring of discrete groups in field extensions and the study of algebraic numbers. Richard Dedekind considered the field ℚ[√2, √3] in this effort. In field theory, extensions are generated by roots of polynomial irreducible over a ground field. Isomorphic fields are associated with permutation groups that move around the roots of the polynomial. This notion, pioneered by Évariste Galois (1830), now forms a standard study Galois theory in mathematics education. In 1936 Ernst Witt published his approach to the quaternion group through Galois theory. A definitive connection was published in 1981, see § Galois group.
Compared to dihedral group.
The quaternion group Q8 has the same order as the dihedral group , but a different structure, as shown by their Cayley and cycle graphs:
In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2. The same cannot be done for Q8, since it has no faithful representation in R2 or R3. D4 can be realized as a subset of the split-quaternions in the same way that Q8 can be viewed as a subset of the quaternions.
Cayley table.
The Cayley table (multiplication table) for Q8 is given by:
Properties.
The elements "i", "j", and "k" all have order four in Q8 and any two of them generate the entire group. Another presentation of Q8 based in only two elements to skip this redundancy is:
$\left \langle x,y \mid x^4 = 1, x^2 = y^2, y^{-1}xy = x^{-1} \right \rangle.$
One may take, for instance, $i = x, j = y,$ and $k = xy$.
The quaternion group has the unusual property of being Hamiltonian: Q8 is non-abelian, but every subgroup is normal. Every Hamiltonian group contains a copy of Q8.
The quaternion group Q8 and the dihedral group D4 are the two smallest examples of a nilpotent non-abelian group.
The center and the commutator subgroup of Q8 is the subgroup $\{e,\bar{e}\}$. The inner automorphism group of Q8 is given by the group modulo its center, i.e. the factor group $\mathrm{Q}_8/\{e,\bar{e}\},$ which is isomorphic to the Klein four-group V. The full automorphism group of Q8 is isomorphic to S4, the symmetric group on four letters (see "Matrix representations" below), and the outer automorphism group of Q8 is thus S4/V, which is isomorphic to S3.
The quaternion group Q8 has five conjugacy classes, $\{e\}, \{\bar{e}\}, \{i,\bar{i}\}, \{j,\bar{j}\}, \{k,\bar{k}\},$ and so five irreducible representations over the complex numbers, with dimensions 1, 1, 1, 1, 2:
Trivial representation.
Sign representations with i, j, k-kernel: Q8 has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup "N", we obtain a one-dimensional representation factoring through the 2-element quotient group "G"/"N". The representation sends elements of "N" to 1, and elements outside "N" to −1.
2-dimensional representation: Described below in "Matrix representations".
The character table of Q8 turns out to be the same as that of D4:
Since the irreducible characters $\chi_\rho$ in the rows above have real values, this gives the decomposition of the real group algebra of $G = \mathrm{Q}_8$ into minimal two-sided ideals:
$\R[\mathrm{Q}_8]=\bigoplus_\rho (e_\rho),$
where the idempotents $e_\rho\in \R[\mathrm{Q}_8]$ correspond to the irreducibles:
$e_\rho = \frac{\dim(\rho)}\sum_{g\in G} \chi_\rho(g^{-1})g,$
so that
$\begin{align}
e_{\text{triv}} &= \tfrac 18(e + \bar e + i +\bar i+j+\bar j+k+\bar k) \\
e_{i\text{-ker}} &= \tfrac 18(e + \bar e + i +\bar i-j-\bar j-k-\bar k) \\
e_{j\text{-ker}} &= \tfrac 18(e + \bar e - i -\bar i+j+\bar j-k-\bar k) \\
e_{k\text{-ker}} &= \tfrac 18(e + \bar e - i -\bar i-j-\bar j+k+\bar k) \\
e_{2} &= \tfrac 28(2e - 2\bar e) = \tfrac 12(e - \bar e)
\end{align}$
Each of these irreducible ideals is isomorphic to a real central simple algebra, the first four to the real field $\R$. The last ideal $(e_2)$ is isomorphic to the skew field of quaternions $\mathbb{H}$ by the correspondence:
$\begin{align}
\tfrac12(e-\bar e) &\longleftrightarrow 1, \\
\tfrac12(i-\bar i) &\longleftrightarrow i, \\
\tfrac12(j-\bar j) &\longleftrightarrow j, \\
\tfrac12(k-\bar k) &\longleftrightarrow k.
\end{align}$
Furthermore, the projection homomorphism $\R[\mathrm{Q}_8]\to (e_2)\cong \mathbb{H}$ given by $r\mapsto re_2$ has kernel ideal generated by the idempotent:
$e_2^\perp = e_1+e_{i\text{-ker}}+e_{j\text{-ker}}+e_{k\text{-ker}} = \frac 12(e+\bar e),$
so the quaternions can also be obtained as the quotient ring $\R[\mathrm{Q}_8]/(e+\bar e)\cong \mathbb H$.
The complex group algebra is thus $\C[\mathrm{Q}_8] \cong \C^{\oplus 4} \oplus M_2(\C),$ where $M_2(\C) \cong \mathbb{H} \otimes_{\R} \C \cong \mathbb{H} \oplus \mathbb{H}$ is the algebra of biquaternions.
Matrix representations.
The two-dimensional irreducible complex representation described above gives the quaternion group Q8 as a subgroup of the general linear group $\operatorname{GL}(2, \C)$. The quaternion group is a multiplicative subgroup of the quaternion algebra:
$\H = \R 1 + \R i + \R j + \R k= \C 1+ \C j,$
which has a regular representation $\rho:\H \to \operatorname{M}(2, \C)$ by left multiplication on itself considered as a complex vector space with basis $\{1,j\},$ so that $z \in \H$ corresponds to the $\C$-linear mapping $\rho_z: a + jb \mapsto z\cdot(a + jb).$ The resulting representation
$\begin{cases} \rho:\mathrm{Q}_8 \to \operatorname{GL}(2,\C)\\ g\longmapsto\rho_g \end{cases}$
is given by:
$\begin{matrix}
1 & 0 \\
0 & 1
\end{pmatrix} &
i & 0 \\
0 & \!\!\!\!-i
\end{pmatrix}&
0 & \!\!\!\!-1 \\
1 & 0
\end{pmatrix}&
0 & \!\!\!\!-i \\
\!\!\!-i & 0
\end{pmatrix} \\
\!\!\!-1 & 0 \\
0 & \!\!\!\!-1
\end{pmatrix} &
\!\!\!-i & 0 \\
0 & i
\end{pmatrix}&
0 & 1 \\
\!\!\!-1 & 0
\end{pmatrix}&
0 & i \\
i & 0
\end{pmatrix}.
$
Since all of the above matrices have unit determinant, this is a representation of Q8 in the special linear group $\operatorname{SL}(2,\C)$.
A variant gives a representation by unitary matrices (table at right). Let $g\in \mathrm{Q}_8$ correspond to the linear mapping $\rho_g:a+bj\mapsto (a + bj)\cdot jg^{-1}j^{-1},$ so that $\rho:\mathrm{Q}_8 \to \operatorname{SU}(2)$ is given by:
$\begin{matrix}
1 & 0 \\
0 & 1
\end{pmatrix} &
i & 0 \\
0 & \!\!\!\!-i
\end{pmatrix}&
0 & 1 \\
\!\!\!-1 & 0
\end{pmatrix}&
0 & i \\
i & 0
\end{pmatrix} \\
\!\!\!-1 & 0 \\
0 & \!\!\!\!-1
\end{pmatrix} &
\!\!\!-i & 0 \\
0 & i
\end{pmatrix}&
0 & \!\!\!\!-1 \\
1 & 0
\end{pmatrix}&
0 & \!\!\!\!-i \\
\!\!\!-i & 0
\end{pmatrix}.
\end{matrix}$
It is worth noting that physicists exclusively use a different convention for the $\operatorname{SU}(2)$ matrix representation to make contact with the usual Pauli matrices:
$\begin{matrix}
1 & 0 \\
0 & 1
\end{pmatrix} = \quad\, 1_{2\times2}
0 & \!\!\!-i\! \\
\!\!-i\!\! & 0
\end{pmatrix} = -i \sigma_x
0 & \!\!\!-1\! \\
1 & 0
\end{pmatrix} = -i \sigma_y
\!\!-i\!\! & 0 \\
0 & i
\end{pmatrix} = -i \sigma_z\\
\!\!-1\! & 0 \\
0 & \!\!\!-1\!
0 & i \\
i & 0
\end{pmatrix} = \,\,\,\, i \sigma_x
0 & 1 \\
\!\!-1\!\! & 0
\end{pmatrix} = \,\,\,\, i \sigma_y
i & 0 \\
0 & \!\!\!-i\!
\end{pmatrix} = \,\,\,\, i \sigma_z.
\end{matrix}$
This particular choice is convenient and elegant when one describes spin-1/2 states in the $(\vec{J}^2, J_z)$ basis and considers angular momentum ladder operators $J_{\pm} = J_x \pm iJ_y.$
There is also an important action of Q8 on the 2-dimensional vector space over the finite field $\mathbb{F}_3 =\{0, 1, -1\}$ (table at right). A modular representation $\rho: \mathrm{Q}_8 \to \operatorname{SL}(2, 3)$ is given by
$\begin{matrix}
1 & 0 \\
0 & 1
\end{pmatrix} &
1 & 1 \\
1 & \!\!\!\!-1
\end{pmatrix} &
\!\!\!-1 & 1 \\
1 & 1
\end{pmatrix} &
0 & \!\!\!\!-1 \\
1 & 0
\end{pmatrix} \\
\!\!\!-1 & 0 \\
0 & \!\!\!\!-1
\end{pmatrix} &
\!\!\!-1 & \!\!\!\!-1 \\
\!\!\!-1 & 1
\end{pmatrix} &
1 & \!\!\!\!-1 \\
\!\!\!-1 & \!\!\!\!-1
\end{pmatrix} &
0 & 1 \\
\!\!\!-1 & 0
\end{pmatrix}.
\end{matrix}$
This representation can be obtained from the extension field:
$ \mathbb{F}_9 = \mathbb{F}_3 [k] = \mathbb{F}_3 1 + \mathbb{F}_3 k,$
where $k^2=-1$ and the multiplicative group $\mathbb{F}_9^{\times}$ has four generators, $\pm(k\pm1),$ of order 8. For each $z \in \mathbb{F}_9,$ the two-dimensional $\mathbb{F}_3$-vector space $\mathbb{F}_9$ admits a linear mapping:
$\begin{cases} \mu_z: \mathbb{F}_9 \to \mathbb{F}_9 \\ \mu_z(a+bk)=z\cdot(a+bk) \end{cases}$
In addition we have the Frobenius automorphism $\phi(a+bk)=(a+bk)^3$ satisfying $\phi^2 = \mu_1 $ and $\phi\mu_z = \mu_{\phi(z)}\phi.$ Then the above representation matrices are:
$\begin{align}
\rho(\bar e) &=\mu_{-1}, \\
\rho(i) &=\mu_{k+1}\phi, \\
\rho(j)&=\mu_{k-1} \phi, \\
\rho(k)&=\mu_{k}.
\end{align}$
This representation realizes Q8 as a normal subgroup of GL(2, 3). Thus, for each matrix $m\in \operatorname{GL}(2,3)$, we have a group automorphism
$\begin{cases} \psi_m:\mathrm{Q}_8\to\mathrm{Q}_8 \\ \psi_m(g)=mgm^{-1} \end{cases}$
with $\psi_I =\psi_{-I}=\mathrm{id}_{\mathrm{Q}_8}.$ In fact, these give the full automorphism group as:
$\operatorname{Aut}(\mathrm{Q}_8) \cong \operatorname{PGL}(2, 3) = \operatorname{GL}(2,3)/\{\pm I\}\cong S_4.$
This is isomorphic to the symmetric group S4 since the linear mappings $m:\mathbb{F}_3^2 \to \mathbb{F}_3^2$ permute the four one-dimensional subspaces of $\mathbb{F}_3^2,$ i.e., the four points of the projective space $\mathbb{P}^1 (\mathbb{F}_3) = \operatorname{PG}(1,3).$
Also, this representation permutes the eight non-zero vectors of $\mathbb{F}_3^2,$ giving an embedding of Q8 in the symmetric group S8, in addition to the embeddings given by the regular representations.
Galois group.
As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field over Q of the polynomial
$x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36$.
The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.
Generalized quaternion group.
A generalized quaternion group Q4"n" of order 4"n" is defined by the presentation
$\langle x,y \mid x^{2n} = y^4 = 1, x^n = y^2, y^{-1}xy = x^{-1}\rangle$
for an integer "n" ≥ 2, with the usual quaternion group given by "n" = 2. Coxeter calls Q4"n" the dicyclic group $\langle 2, 2, n\rangle$, a special case of the binary polyhedral group $\langle \ell, m, n\rangle$ and related to the polyhedral group $(p,q,r)$ and the dihedral group $(2,2,n)$. The generalized quaternion group can be realized as the subgroup of $\operatorname{GL}_2(\Complex)$ generated by
$\left(\begin{array}{cc}
\omega_n & 0 \\
0 & \overline{\omega}_n
\right)
0 & -1 \\
1 & 0
\right)
$
where $\omega_n = e^{i\pi/n}$. It can also be realized as the subgroup of unit quaternions generated by $x=e^{i\pi/n}$ and $y=j$.
The generalized quaternion groups have the property that every abelian subgroup is cyclic. It can be shown that a finite "p"-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. Another characterization is that a finite "p"-group in which there is a unique subgroup of order "p" is either cyclic or a 2-group isomorphic to generalized quaternion group. In particular, for a finite field "F" with odd characteristic, the 2-Sylow subgroup of SL2("F") is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, . Letting "pr" be the size of "F", where "p" is prime, the size of the 2-Sylow subgroup of SL2("F") is 2"n", where "n" = ord2("p"2 − 1) + ord2("r").
The Brauer–Suzuki theorem shows that the groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.
Another terminology reserves the name "generalized quaternion group" for a dicyclic group of order a power of 2, which admits the presentation
$\langle x,y \mid x^{2^m} = y^4 = 1, x^{2^{m-1}} = y^2, y^{-1}xy = x^{-1}\rangle.$
Notes.
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In mathematics, specifically group theory, a Hall subgroup of a finite group "G" is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall (1928).
Definitions.
A Hall divisor (also called a unitary divisor) of an integer "n" is a divisor "d" of "n" such that
"d" and "n"/"d" are coprime. The easiest way to find the Hall divisors is to write the prime power factorization of the number in question and take any subset of the factors. For example, to find the Hall divisors of 60, its prime power factorization is 22 × 3 × 5, so one takes any product of 3, 22 = 4, and 5. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60.
A Hall subgroup of "G" is a subgroup whose order is a Hall divisor of the order of "G". In other words, it is a subgroup whose order is coprime to its index.
If "π" is a set of primes, then a Hall "π"-subgroup is a subgroup whose order is a product of primes in "π", and whose index is not divisible by any primes in "π".
Hall's theorem.
proved that if "G" is a finite solvable group and "π"
is any set of primes, then "G" has a Hall "π"-subgroup, and any two Hall "π"-subgroups are conjugate. Moreover, any subgroup whose order is
a product of primes in "π" is contained in some Hall "π"-subgroup. This result can be thought of as a generalization of Sylow's Theorem to Hall subgroups, but the examples above show that such a generalization is false when the group is not solvable.
The existence of Hall subgroups can be proved by induction on the order of "G", using the fact that every finite solvable group has a normal elementary abelian subgroup. More precisely, fix a minimal normal subgroup "A", which is either a "π"-group or a "π′"-group as "G" is "π"-separable. By induction there is a subgroup "H" of "G" containing "A" such that "H"/"A" is a Hall "π"-subgroup of "G"/"A". If "A" is a "π"-group then "H" is a Hall "π"-subgroup of "G". On the other hand, if "A" is a "π′"-group, then by the Schur–Zassenhaus theorem "A" has a complement in "H", which is a Hall "π"-subgroup of "G".
A converse to Hall's theorem.
Any finite group that has a Hall "π"-subgroup for every set of primes "π" is solvable. This is a generalization of Burnside's theorem that any group whose order is of the form "paqb" for primes "p" and "q" is solvable, because Sylow's theorem implies that all Hall subgroups exist. This does not (at present) give another proof of Burnside's theorem, because Burnside's theorem is used to prove this converse.
Sylow systems.
A Sylow system is a set of Sylow "p"-subgroups "Sp" for each prime "p" such that "SpSq" = "SqSp" for all "p" and "q". If we have a Sylow system, then the subgroup generated by the groups "Sp" for "p" in "π" is a Hall "π"-subgroup. A more precise version of Hall's theorem says that any solvable group has a Sylow system, and any two Sylow systems are conjugate.
Normal Hall subgroups.
Any normal Hall subgroup "H" of a finite group "G" possesses a complement, that is, there is some subgroup "K" of "G" that intersects "H" trivially and such that "HK" = "G" (so "G" is a semidirect product of "H" and "K"). This is the Schur–Zassenhaus theorem.
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In mathematics, specifically group theory, the free product is an operation that takes two groups "G" and "H" and constructs a new group "G" ∗ "H". The result contains both "G" and "H" as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from "G" and "H" into a group "K" factor uniquely through a homomorphism from "G" ∗ "H" to "K". Unless one of the groups "G" and "H" is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the universal group with a given set of generators).
The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. Therefore, the free product is not the coproduct in the category of abelian groups.
The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological spaces whose intersection is also path-connected is always an amalgamated free product of the fundamental groups of the spaces. In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is, under certain conditions given in the Seifert van-Kampen theorem, the free product of the fundamental groups of the spaces.
Free products are also important in Bass–Serre theory, the study of groups acting by automorphisms on trees. Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extensions. Using the action of the modular group on a certain tessellation of the hyperbolic plane, it follows from this theory that the modular group is isomorphic to the free product of cyclic groups of orders 4 and 6 amalgamated over a cyclic group of order 2.
Construction.
If "G" and "H" are groups, a word in "G" and "H" is a product of the form
$s_1 s_2 \cdots s_n,$
where each "s""i" is either an element of "G" or an element of "H". Such a word may be reduced using the following operations:
Every reduced word is an alternating product of elements of "G" and elements of "H", e.g.
$g_1 h_1 g_2 h_2 \cdots g_k h_k.$
The free product "G" ∗ "H" is the group whose elements are the reduced words in "G" and "H", under the operation of concatenation followed by reduction.
For example, if "G" is the infinite cyclic group $\langle x\rangle$, and "H" is the infinite cyclic group $\langle y\rangle$, then every element of "G" ∗ "H" is an alternating product of powers of "x" with powers of "y". In this case, "G" ∗ "H" is isomorphic to the free group generated by "x" and "y".
Presentation.
Suppose that
$G = \langle S_G \mid R_G \rangle$
is a presentation for "G" (where "S""G" is a set of generators and "R""G" is a set of relations), and suppose that
$H = \langle S_H \mid R_H \rangle$
is a presentation for "H". Then
$G * H = \langle S_G \cup S_H \mid R_G \cup R_H \rangle.$
That is, "G" ∗ "H" is generated by the generators for "G" together with the generators for "H", with relations consisting of the relations from "G" together with the relations from "H" (assume here no notational clashes so that these are in fact disjoint unions).
Examples.
For example, suppose that "G" is a cyclic group of order 4,
$G = \langle x \mid x^4 = 1 \rangle,$
and "H" is a cyclic group of order 5
$H = \langle y \mid y^5 = 1 \rangle.$
Then "G" ∗ "H" is the infinite group
$G * H = \langle x, y \mid x^4 = y^5 = 1 \rangle.$
Because there are no relations in a free group, the free product of free groups is always a free group. In particular,
$F_m * F_n \cong F_{m+n},$
where "F""n" denotes the free group on "n" generators.
Another example is the modular group $PSL_2(\mathbf Z)$. It is isomorphic to the free product of two cyclic groups:
$PSL_2(\mathbf Z) \cong (\mathbf Z / 2 \mathbf Z) \ast (\mathbf Z / 3 \mathbf Z).$
Generalization: Free product with amalgamation.
The more general construction of free product with amalgamation is correspondingly a special kind of pushout in the same category. Suppose $G$ and $H$ are given as before, along with monomorphisms (i.e. injective group homomorphisms):
$\varphi : F \rightarrow G \ \, $ and $\ \, \psi : F \rightarrow H,$
where $F$ is some arbitrary group. Start with the free product $G * H$ and adjoin as relations
$\varphi(f)\psi(f)^{-1}=1$
for every $f$ in $F$. In other words, take the smallest normal subgroup $N$ of $G * H$ containing all elements on the left-hand side of the above equation, which are tacitly being considered in $G * H$ by means of the inclusions of $G$ and $H$ in their free product. The free product with amalgamation of $G$ and $H$, with respect to $\varphi$ and $\psi$, is the quotient group
$(G * H)/N.\,$
The amalgamation has forced an identification between $ \varphi(F) $ in $G$ with $\psi(F)$ in $H$, element by element. This is the construction needed to compute the fundamental group of two connected spaces joined along a path-connected subspace, with $F$ taking the role of the fundamental group of the subspace. See: Seifert–van Kampen theorem.
Karrass and Solitar have given a description of the subgroups of a free product with amalgamation. For example, the homomorphisms from $G$ and $H$ to the quotient group $(G * H)/N$ that are induced by $\varphi$ and $\psi$ are both injective, as is the induced homomorphism from $F$.
Free products with amalgamation and a closely related notion of HNN extension are basic building blocks in Bass–Serre theory of groups acting on trees.
In other branches.
One may similarly define free products of other algebraic structures than groups, including algebras over a field. Free products of algebras of random variables play the same role in defining "freeness" in the theory of free probability that Cartesian products play in defining statistical independence in classical probability theory.
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Studies linear representations of finite groups over a field K of positive characteristic p
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field "K" of positive characteristic "p", necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.
Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups were too small in an appropriate sense. Also, a general result on embedding of elements of order 2 in finite groups called the Z* theorem, proved by George Glauberman using the theory developed by Brauer, was particularly useful in the classification program.
If the characteristic "p" of "K" does not divide the order |"G"|, then modular representations are completely reducible, as with "ordinary" (characteristic 0) representations, by virtue of Maschke's theorem. In the other case, when |"G"| ≡ 0 mod "p", the process of averaging over the group needed to prove Maschke's theorem breaks down, and representations need not be completely reducible. Much of the discussion below implicitly assumes that the field "K" is sufficiently large (for example, "K" algebraically closed suffices), otherwise some statements need refinement.
History.
The earliest work on representation theory over finite fields is by who showed that when "p" does not divide the order of the group, the representation theory is similar to that in characteristic 0. He also investigated modular invariants of some finite groups. The systematic study of modular representations, when the characteristic "p" divides the order of the group, was started by and was continued by him for the next few decades.
Example.
Finding a representation of the cyclic group of two elements over F2 is equivalent to the problem of finding matrices whose square is the identity matrix. Over every field of characteristic other than 2, there is always a basis such that the matrix can be written as a diagonal matrix with only 1 or −1 occurring on the diagonal, such as
$
1 & 0\\
0 & -1
\end{bmatrix}.
$
Over F2, there are many other possible matrices, such as
$
1 & 1\\
0 & 1
\end{bmatrix}.
$
Over an algebraically closed field of positive characteristic, the representation theory of a finite cyclic group is fully explained by the theory of the Jordan normal form. Non-diagonal Jordan forms occur when the characteristic divides the order of the group.
Ring theory interpretation.
Given a field "K" and a finite group "G", the group algebra "K"["G"] (which is the "K"-vector space with "K"-basis consisting of the elements of "G", endowed with algebra multiplication by extending the multiplication of "G" by linearity) is an Artinian ring.
When the order of "G" is divisible by the characteristic of "K", the group algebra is not semisimple, hence has non-zero Jacobson radical. In that case, there are finite-dimensional modules for the group algebra that are not projective modules. By contrast, in the characteristic 0 case every irreducible representation is a direct summand of the regular representation, hence is projective.
Brauer characters.
Modular representation theory was developed by Richard Brauer from about 1940 onwards to study in greater depth the relationships between the
characteristic "p" representation theory, ordinary character theory and structure of "G", especially as the latter relates to the embedding of, and relationships between, its "p"-subgroups. Such results can be applied in group theory to problems not directly phrased in terms of representations.
Brauer introduced the notion now known as the Brauer character. When "K" is algebraically closed of positive characteristic "p", there is a bijection between roots of unity in "K" and complex roots of unity of order prime to "p". Once a choice of such a bijection is fixed, the Brauer character of a representation assigns to each group element of order coprime to "p" the sum of complex roots of unity corresponding to the eigenvalues (including multiplicities) of that element in the given representation.
The Brauer character of a representation determines its composition
factors but not, in general, its equivalence type. The irreducible
Brauer characters are those afforded by the simple modules.
These are integral (though not necessarily non-negative) combinations
of the restrictions to elements of order coprime to "p" of the ordinary irreducible
characters. Conversely, the restriction to the elements of order coprime to "p" of
each ordinary irreducible character is uniquely expressible as a non-negative
integer combination of irreducible Brauer characters.
Reduction (mod "p").
In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering the
group algebra of the group "G" over a complete discrete
valuation ring "R" with residue field "K" of positive
characteristic "p" and field of fractions "F" of characteristic
0, such as the "p"-adic integers. The structure of "R"["G"] is closely related both to
the structure of the group algebra "K"["G"] and to the structure of the semisimple group algebra "F"["G"], and there is much interplay
between the module theory of the three algebras.
Each "R"["G"]-module naturally gives rise to an "F"["G"]-module,
and, by a process often known informally as reduction (mod "p"),
to a "K"["G"]-module. On the other hand, since "R" is a
principal ideal domain, each finite-dimensional "F"["G"]-module
arises by extension of scalars from an "R"["G"]-module. In general,
however, not all "K"["G"]-modules arise as reductions (mod "p") of
"R"["G"]-modules. Those that do are liftable.
Number of simple modules.
In ordinary representation theory, the number of simple modules "k"("G") is equal to the number of conjugacy classes of "G". In the modular case, the number "l"("G") of simple modules is equal to the number of conjugacy classes whose elements have order coprime to the relevant prime "p", the so-called "p"-regular classes.
Blocks and the structure of the group algebra.
In modular representation theory, while Maschke's theorem does not hold
when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as blocks. When the field "F" has characteristic 0, or characteristic coprime to the group order, there is still such a decomposition of the group algebra "F"["G"] as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent when "F" is sufficiently large: each block is a full matrix algebra over "F", the endomorphism ring of the vector space underlying the associated simple module.
To obtain the blocks, the identity element of the group "G" is decomposed as a sum of primitive idempotents
in "Z"("R"[G]), the center of the group algebra over the maximal order "R" of "F". The block corresponding to the primitive idempotent
"e" is the two-sided ideal "e" "R"["G"]. For each indecomposable "R"["G"]-module, there is only one such primitive idempotent that does not annihilate it, and the module is said to belong to (or to be in) the corresponding block (in which case, all its composition factors also belong to that block). In particular, each simple module belongs to a unique block. Each ordinary irreducible character may also be assigned to a unique block according to its decomposition as a sum of irreducible Brauer characters. The block containing the trivial module is known as the principal block.
Projective modules.
In ordinary representation theory, every indecomposable module is irreducible, and so every module is projective. However, the simple modules with characteristic dividing the group order are rarely projective. Indeed, if a simple module is projective, then it is the only simple module in its block, which is then isomorphic to the endomorphism algebra of the underlying vector space, a full matrix algebra. In that case, the block is said to have 'defect 0'. Generally, the structure of projective modules is difficult to determine.
For the group algebra of a finite group, the (isomorphism types of) projective indecomposable modules are in a one-to-one correspondence with the (isomorphism types of) simple modules: the socle of each projective indecomposable is simple (and isomorphic to the top), and this affords the bijection, as non-isomorphic projective indecomposables have
non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the group algebra (viewed as the regular module) is the dimension of its socle (for large enough fields of characteristic zero, this recovers the fact that each simple module occurs with multiplicity equal to its dimension as a direct summand of the regular module).
Each projective indecomposable module (and hence each projective module) in positive characteristic "p" may be lifted to a module in characteristic 0. Using the ring "R" as above, with residue field "K", the identity element of "G" may be decomposed as a sum of mutually orthogonal primitive idempotents (not necessarily
central) of "K"["G"]. Each projective indecomposable "K"["G"]-module is isomorphic to "e"."K"["G"] for a primitive idempotent "e" that occurs in this decomposition. The idempotent "e" lifts to a primitive idempotent, say "E", of "R"["G"], and the left module "E"."R"["G"] has reduction (mod "p") isomorphic to "e"."K"["G"].
Some orthogonality relations for Brauer characters.
When a projective module is lifted, the associated character vanishes on all elements of order divisible by "p", and (with consistent choice of roots of unity), agrees with the Brauer character of the original characteristic "p" module on "p"-regular elements. The (usual character-ring) inner product of the Brauer character of a projective indecomposable with any other Brauer character can thus be defined: this is 0 if the
second Brauer character is that of the socle of a non-isomorphic projective indecomposable, and 1
if the second Brauer character is that of its own socle. The multiplicity of an ordinary irreducible
character in the character of the lift of a projective indecomposable is equal to the number
of occurrences of the Brauer character of the socle of the projective indecomposable when the restriction of the ordinary character to "p"-regular elements is expressed as a sum of irreducible Brauer characters.
Decomposition matrix and Cartan matrix.
The composition factors of the projective indecomposable modules may be calculated as follows:
Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, the irreducible ordinary characters may be decomposed as non-negative integer combinations of the irreducible Brauer characters. The integers involved can be placed in a matrix, with the ordinary irreducible characters assigned rows and the irreducible Brauer characters assigned columns. This is referred to as the "decomposition matrix", and is frequently labelled "D". It is customary to place the trivial ordinary and Brauer characters in the first row and column respectively. The product of the transpose of "D" with "D" itself
results in the Cartan matrix, usually denoted "C"; this is a symmetric matrix such that the entries in its "j"-th row are the multiplicities of the respective simple modules as composition
factors of the "j"-th projective indecomposable module. The Cartan
matrix is non-singular; in fact, its determinant is a power of the
characteristic of "K".
Since a projective indecomposable module in a given block has
all its composition factors in that same block, each block has
its own Cartan matrix.
Defect groups.
To each block "B" of the group algebra "K"["G"], Brauer associated a certain "p"-subgroup, known as its defect group (where "p" is the characteristic of "K"). Formally, it is the largest "p"-subgroup
"D" of "G" for which there is a Brauer correspondent of "B" for the
subgroup $DC_G(D)$, where $C_G(D)$ is the centralizer of "D" in "G".
The defect group of a block is unique up to conjugacy and has a strong influence on the structure of the block. For example, if the defect group is trivial, then the block contains just one simple module, just one ordinary character, the ordinary and Brauer irreducible characters agree on elements of order prime to the relevant characteristic "p", and the simple module is projective. At the other extreme, when "K" has characteristic "p", the Sylow "p"-subgroup of the finite group "G" is a defect group for the principal block of "K"["G"].
The order of the defect group of a block has many arithmetical characterizations related to representation theory. It is the largest invariant factor of the Cartan matrix of the block, and occurs with
multiplicity one. Also, the power of "p" dividing the index of the defect group of a block is the greatest common divisor of the powers of "p" dividing the dimensions of the simple modules in that block, and this coincides with the greatest common divisor of the powers of "p" dividing the degrees of the ordinary irreducible characters in that block.
Other relationships between the defect group of a block and character theory include Brauer's result that if no conjugate of the "p"-part of a group element "g" is in the defect group of a given block, then each irreducible character in that block vanishes at "g". This is one of many consequences of Brauer's second main theorem.
The defect group of a block also has several characterizations in the more module-theoretic approach to block theory, building on the work of J. A. Green, which associates a "p"-subgroup
known as the vertex to an indecomposable module, defined in terms of relative projectivity of the module. For example, the vertex of each indecomposable module in a block is contained (up to conjugacy)
in the defect group of the block, and no proper subgroup of the defect group has that property.
Brauer's first main theorem states that the number of blocks of a finite group that have a given "p"-subgroup as defect group is the same as the corresponding number for the normalizer in the group of that "p"-subgroup.
The easiest block structure to analyse with non-trivial defect group is when the latter is cyclic. Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer, E.C. Dade, J.A. Green and J.G. Thompson, among others. In all other cases, there are infinitely many isomorphism types of indecomposable modules in the block.
Blocks whose defect groups are not cyclic can be divided into two types: tame and wild. The tame blocks (which only occur for the prime 2) have as a defect group a dihedral group, semidihedral group or (generalized) quaternion group, and their structure has been broadly determined in a series of papers by Karin Erdmann. The indecomposable modules in wild blocks are extremely difficult to classify, even in principle.
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Problem a computer might be able to solve
In theoretical computer science, a computational problem is a problem that may be solved by an algorithm. For example, the problem of factoring
"Given a positive integer "n", find a nontrivial prime factor of "n"."
is a computational problem. A computational problem can be viewed as a set of "instances" or "cases" together with a, possibly empty, set of "solutions" for every instance/case. For example, in the factoring problem, the instances are the integers "n", and solutions are prime numbers "p" that are the nontrivial prime factors of "n".
Computational problems are one of the main objects of study in theoretical computer science. The field of computational complexity theory attempts to determine the amount of resources (computational complexity) solving a given problem will require and explain why some problems are intractable or undecidable. Computational problems belong to complexity classes that define broadly the resources (e.g. time, space/memory, energy, circuit depth) it takes to compute (solve) them with various abstract machines. For example, the complexity classes
Both instances and solutions are represented by binary strings, namely elements of {0, 1}*. For example, natural numbers are usually represented as binary strings using binary encoding. This is important since the complexity is expressed as a function of the length of the input representation.
Types.
Decision problem.
A decision problem is a computational problem where the answer for every instance is either yes or no. An example of a decision problem is "primality testing":
"Given a positive integer "n", determine if "n" is prime."
A decision problem is typically represented as the set of all instances for which the answer is "yes". For example, primality testing can be represented as the infinite set
"L" = {2, 3, 5, 7, 11, ...}
Search problem.
In a search problem, the answers can be arbitrary strings. For example, factoring is a search problem where the instances are (string representations of) positive integers and the solutions are (string representations of) collections of primes.
A search problem is represented as a relation consisting of all the instance-solution pairs, called a "search relation". For example, factoring can be represented as the relation
"R" = {(4, 2), (6, 2), (6, 3), (8, 2), (9, 3), (10, 2), (10, 5)...}
which consist of all pairs of numbers ("n", "p"), where "p" is a nontrivial prime factor of "n".
Counting problem.
A counting problem asks for the number of solutions to a given search problem. For example, a counting problem associated with factoring is
"Given a positive integer "n", count the number of nontrivial prime factors of "n"."
A counting problem can be represented by a function "f" from {0, 1}* to the nonnegative integers. For a search relation "R", the counting problem associated to "R" is the function
"fR"(x) = |{"y": "R"("x", "y") }|.
Optimization problem.
An optimization problem asks for finding a "best possible" solution among the set of all possible solutions to a search problem. One example is the "maximum independent set" problem:
"Given a graph "G", find an independent set of "G" of maximum size."
Optimization problems are represented by their objective function and their constraints.
Function problem.
In a function problem a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem, that is, it isn't just "yes" or "no". One of the most famous examples is the "traveling salesman" problem:
"Given a list of cities and the distances between each pair of cities, find the shortest possible route that visits each city exactly once and returns to the origin city."
It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.
Promise problem.
In computational complexity theory, it is usually implicitly assumed that any string in {0, 1}* represents an instance of the computational problem in question. However, sometimes not all strings {0, 1}* represent valid instances, and one specifies a proper subset of {0, 1}* as the set of "valid instances". Computational problems of this type are called promise problems.
The following is an example of a (decision) promise problem:
"Given a graph "G", determine if every independent set in "G" has size at most 5, or "G" has an independent set of size at least 10."
Here, the valid instances are those graphs whose maximum independent set size is either at most 5 or at least 10.
Decision promise problems are usually represented as pairs of disjoint subsets ("L"yes, "L"no) of {0, 1}*. The valid instances are those in "L"yes ∪ "L"no.
"L"yes and "L"no represent the instances whose answer is "yes" and "no", respectively.
Promise problems play an important role in several areas of computational complexity, including hardness of approximation, property testing, and interactive proof systems.
Notes.
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abstract_algebra
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Algorithm for integer factorization
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra.
Practically speaking, ECM is considered a special-purpose factoring algorithm, as it is most suitable for finding small factors. Currently[ [update]], it is still the best algorithm for divisors not exceeding 50 to 60 digits, as its running time is dominated by the size of the smallest factor "p" rather than by the size of the number "n" to be factored. Frequently, ECM is used to remove small factors from a very large integer with many factors; if the remaining integer is still composite, then it has only large factors and is factored using general-purpose techniques. The largest factor found using ECM so far has 83 decimal digits and was discovered on 7 September 2013 by R. Propper. Increasing the number of curves tested improves the chances of finding a factor, but they are not linear with the increase in the number of digits.
Algorithm.
The Lenstra elliptic-curve factorization method to find a factor of a given natural number $n$ works as follows:
The time complexity depends on the size of the number's smallest prime factor and can be represented by exp[(√2 + o(1)) √ln "p" ln ln "p"], where "p" is the smallest factor of "n", or $L_p\left[\frac{1}{2},\sqrt{2}\right]$, in L-notation.
Explanation.
If "p" and "q" are two prime divisors of "n", then "y"2 = "x"3 + "ax" + "b" (mod "n") implies the same equation also modulo "p" and modulo "q". These two smaller elliptic curves with the $\boxplus$-addition are now genuine groups. If these groups have "N""p" and "Nq" elements, respectively, then for any point "P" on the original curve, by Lagrange's theorem, "k" > 0 is minimal such that $kP=\infty$ on the curve modulo "p" implies that "k" divides "N""p"; moreover, $N_p P=\infty$. The analogous statement holds for the curve modulo "q". When the elliptic curve is chosen randomly, then "N""p" and "N""q" are random numbers close to "p" + 1 and "q" + 1, respectively (see below). Hence it is unlikely that most of the prime factors of "N""p" and "N""q" are the same, and it is quite likely that while computing "eP", we will encounter some "kP" that is ∞ modulo "p" but not modulo "q", or vice versa. When this is the case, "kP" does not exist on the original curve, and in the computations we found some "v" with either gcd("v","p") = "p" or gcd("v", "q") = "q", but not both. That is, gcd("v", "n") gave a non-trivial factor of "n".
ECM is at its core an improvement of the older "p" − 1 algorithm. The "p" − 1 algorithm finds prime factors "p" such that "p" − 1 is b-powersmooth for small values of "b". For any "e", a multiple of "p" − 1, and any "a" relatively prime to "p", by Fermat's little theorem we have "a""e" ≡ 1 (mod "p"). Then gcd("a""e" − 1, "n") is likely to produce a factor of "n". However, the algorithm fails when "p" - 1 has large prime factors, as is the case for numbers containing strong primes, for example.
ECM gets around this obstacle by considering the group of a random elliptic curve over the finite field Z"p", rather than considering the multiplicative group of Z"p" which always has order "p" − 1.
The order of the group of an elliptic curve over Z"p" varies (quite randomly) between "p" + 1 − 2√"p" and "p" + 1 + 2√"p" by Hasse's theorem, and is likely to be smooth for some elliptic curves. Although there is no proof that a smooth group order will be found in the Hasse-interval, by using heuristic probabilistic methods, the Canfield–Erdős–Pomerance theorem with suitably optimized parameter choices, and the L-notation, we can expect to try L[√2/2, √2] curves before getting a smooth group order. This heuristic estimate is very reliable in practice.
Example usage.
The following example is from , with some details added.
We want to factor "n" = 455839. Let's choose the elliptic curve "y"2 = "x"3 + 5"x" – 5, with the point "P" = (1, 1) on it, and let's try to compute (10!)"P".
The slope of the tangent line at some point "A"=("x", "y") is "s" = (3"x"2 + 5)/(2"y") (mod n). Using "s" we can compute 2"A". If the value of "s" is of the form "a/b" where "b" > 1 and gcd("a","b") = 1, we have to find the modular inverse of "b". If it does not exist, gcd("n","b") is a non-trivial factor of "n".
First we compute 2"P". We have "s"("P") = "s"(1,1) = 4, so the coordinates of 2"P" = ("x′", "y′") are "x′" = "s"2 – 2"x" = 14 and "y′" = "s"("x" – "x′") – "y" = 4(1 – 14) – 1 = –53, all numbers understood (mod "n"). Just to check that this 2"P" is indeed on the curve: (–53)2 = 2809 = 143 + 5·14 – 5.
Then we compute 3(2"P"). We have "s"(2"P") = "s"(14,-53) = –593/106 (mod "n"). Using the Euclidean algorithm: 455839 = 4300·106 + 39, then 106 = 2·39 + 28, then 39 = 28 + 11, then 28 = 2·11 + 6, then 11 = 6 + 5, then 6 = 5 + 1. Hence gcd(455839, 106) = 1, and working backwards (a version of the extended Euclidean algorithm): 1 = 6 – 5 = 2·6 – 11 = 2·28 – 5·11 = 7·28 – 5·39 = 7·106 – 19·39 = 81707·106 – 19·455839. Hence 106−1 = 81707 (mod 455839), and –593/106 = –133317 (mod 455839). Given this "s", we can compute the coordinates of 2(2"P"), just as we did above: 4"P" = (259851, 116255). Just to check that this is indeed a point on the curve: "y"2 = 54514 = "x"3 + 5"x" – 5 (mod 455839). After this, we can compute $3(2P) = 4P \boxplus 2P$.
We can similarly compute 4!"P", and so on, but 8!"P" requires inverting 599 (mod 455839). The Euclidean algorithm gives that 455839 is divisible by 599, and we have found a factorization 455839 = 599·761.
The reason that this worked is that the curve (mod 599) has 640 = 27·5 points, while (mod 761) it has 777 = 3·7·37 points. Moreover, 640 and 777 are the smallest positive integers "k" such that "kP" = ∞ on the curve (mod 599) and (mod 761), respectively. Since 8! is a multiple of 640 but not a multiple of 777, we have 8!"P" = ∞ on the curve (mod 599), but not on the curve (mod 761), hence the repeated addition broke down here, yielding the factorization.
The algorithm with projective coordinates.
Before considering the projective plane over $(\Z/n\Z)/\sim,$ first consider a 'normal' projective space over ℝ: Instead of points, lines through the origin are studied. A line may be represented as a non-zero point $(x,y,z)$, under an equivalence relation ~ given by: $(x,y,z)\sim(x',y',z')$ ⇔ ∃ c ≠ 0 such that "x' = cx", "y' = cy" and "z' = cz". Under this equivalence relation, the space is called the projective plane $\mathbb{P}^2$; points, denoted by $(x:y:z)$, correspond to lines in a three-dimensional space that pass through the origin. Note that the point $(0:0:0)$ does not exist in this space since to draw a line in any possible direction requires at least one of x',y' or z' ≠ 0. Now observe that almost all lines go through any given reference plane - such as the ("X","Y",1)-plane, whilst the lines precisely parallel to this plane, having coordinates ("X,Y",0), specify directions uniquely, as 'points at infinity' that are used in the affine ("X,Y")-plane it lies above.
In the algorithm, only the group structure of an elliptic curve over the field ℝ is used. Since we do not necessarily need the field ℝ, a finite field will also provide a group structure on an elliptic curve. However, considering the same curve and operation over $(\Z/n\Z)/\sim$ with n not a prime does not give a group. The Elliptic Curve Method makes use of the failure cases of the addition law.
We now state the algorithm in projective coordinates. The neutral element is then given by the point at infinity $(0:1:0)$. Let n be a (positive) integer and consider the elliptic curve (a set of points with some structure on it) $E(\Z/n\Z)=\{(x:y:z) \in \mathbb{P}^2\ |\ y^2z=x^3+axz^2+bz^3\}$.
In point 5 it is said that under the right circumstances a non-trivial divisor can be found. As pointed out in Lenstra's article (Factoring Integers with Elliptic Curves) the addition needs the assumption $\gcd(x_1-x_2,n)=1$. If $P,Q$ are not $(0:1:0)$ and distinct (otherwise addition works similarly, but is a little different), then addition works as follows:
If addition fails, this will be due to a failure calculating $\lambda.$ In particular, because $(x_1-x_2)^{-1}$ can not always be calculated if n is not prime (and therefore $\Z/n\Z$ is not a field). Without making use of $\Z/n\Z$ being a field, one could calculate:
This calculation is always legal and if the gcd of the Z-coordinate with n ≠ (1 or n), so when simplifying fails, a non-trivial divisor of n is found.
Twisted Edwards curves.
The use of Edwards curves needs fewer modular multiplications and less time than the use of Montgomery curves or Weierstrass curves (other used methods). Using Edwards curves you can also find more primes.
Definition. Let $k$ be a field in which $2 \neq 0$, and let $a,d \in k\setminus\{0\}$ with $a\neq d$. Then the twisted Edwards curve $E_{E,a,d}$ is given by $ax^2+y^2=1+dx^2y^2.$ An Edwards curve is a twisted Edwards curve in which $a=1$.
There are five known ways to build a set of points on an Edwards curve: the set of affine points, the set of projective points, the set of inverted points, the set of extended points and the set of completed points.
The set of affine points is given by:
$\{(x,y)\in \mathbb{A}^2 : ax^2+y^2=1+dx^2y^2\}$.
The addition law is given by
$(e,f),(g,h) \mapsto \left(\frac{eh+fg}{1+ degfh},\frac{fh-aeg}{1-degfh}\right).$
The point (0,1) is its neutral element and the inverse of $(e,f)$ is $(-e,f)$.
The other representations are defined similar to how the projective Weierstrass curve follows from the affine.
Any elliptic curve in Edwards form has a point of order 4. So the torsion group of an Edwards curve over $\Q$ is isomorphic to either $\Z/4\Z, \Z/8\Z, \Z/12\Z,\Z/2\Z \times \Z/4\Z$ or $\Z/2\Z\times \Z/8\Z$.
The most interesting cases for ECM are $\Z/12\Z$ and $\Z/2\Z\times \Z/8\Z$, since they force the group orders of the curve modulo primes to be divisible by 12 and 16 respectively. The following curves have a torsion group isomorphic to $\Z/12\Z$:
Every Edwards curve with a point of order 3 can be written in the ways shown above. Curves with torsion group isomorphic to $\Z/2\Z\times \Z/8\Z$ and $\Z/2\Z\times \Z/4\Z$ may be more efficient at finding primes.
Stage 2.
The above text is about the first stage of elliptic curve factorisation. There one hopes to find a prime divisor p such that $sP$ is the neutral element of $E(\mathbb{Z}/p\mathbb{Z})$.
In the second stage one hopes to have found a prime divisor q such that $sP$ has small prime order in $E(\mathbb{Z}/q\mathbb{Z})$.
We hope the order to be between $B_1$ and $B_2$, where $B_1$ is determined in stage 1 and $B_2$ is new stage 2 parameter.
Checking for a small order of $sP$, can be done by computing $(ls)P$ modulo n for each prime l.
GMP-ECM and EECM-MPFQ.
The use of Twisted Edwards elliptic curves, as well as other techniques were used by Bernstein et al to provide an optimized implementation of ECM. Its only drawback is that it works on smaller composite numbers than the more general purpose implementation, GMP-ECM of Zimmerman.
Hyperelliptic-curve method (HECM).
There are recent developments in using hyperelliptic curves to factor integers. Cosset shows in his article (of 2010) that one can build a hyperelliptic curve with genus two (so a curve $y^2 = f(x)$ with f of degree 5), which gives the same result as using two "normal" elliptic curves at the same time. By making use of the Kummer surface, calculation is more efficient. The disadvantages of the hyperelliptic curve (versus an elliptic curve) are compensated by this alternative way of calculating. Therefore, Cosset roughly claims that using hyperelliptic curves for factorization is no worse than using elliptic curves.
Quantum version (GEECM).
Bernstein, Heninger, Lou, and Valenta suggest GEECM, a quantum version of ECM with Edwards curves. It uses Grover's algorithm to roughly double the length of the primes found compared to standard EECM, assuming a quantum computer with sufficiently many qubits and of comparable speed to the classical computer running EECM.
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88699
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abstract_algebra
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Unsolved problem in mathematics:
Is every finite group the Galois group of a Galois extension of the rational numbers?
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers $\mathbb{Q}$. This problem, first posed in the early 19th century, is unsolved.
There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of $\mathbb{Q}$ having a particular group as Galois group. These groups include all of degree no greater than 5. There also are groups known not to have generic polynomials, such as the cyclic group of order 8.
More generally, let G be a given finite group, and let K be a field. Then the question is this: is there a Galois extension field "L"/"K" such that the Galois group of the extension is isomorphic to G? One says that G is realizable over K if such a field L exists.
Partial results.
There is a great deal of detailed information in particular cases. It is known that every finite group is realizable over any function field in one variable over the complex numbers $\mathbb{C}$, and more generally over function fields in one variable over any algebraically closed field of characteristic zero. Igor Shafarevich showed that every finite solvable group is realizable over $\mathbb{Q}$. It is also known that every sporadic group, except possibly the Mathieu group "M"23, is realizable over $\mathbb{Q}$.
David Hilbert had shown that this question is related to a rationality question for G:
If K is any extension of $\mathbb{Q}$, on which G acts as an automorphism group and the invariant field "KG" is rational over $\mathbb{Q}$, then G is realizable over $\mathbb{Q}$.
Here "rational" means that it is a purely transcendental extension of $\mathbb{Q}$, generated by an algebraically independent set. This criterion can for example be used to show that all the symmetric groups are realizable.
Much detailed work has been carried out on the question, which is in no sense solved in general. Some of this is based on constructing G geometrically as a Galois covering of the projective line: in algebraic terms, starting with an extension of the field $\mathbb{Q}(t)$ of rational functions in an indeterminate t. After that, one applies Hilbert's irreducibility theorem to specialise t, in such a way as to preserve the Galois group.
All permutation groups of degree 16 or less are known to be realizable over $\mathbb{Q}$; the group PSL(2,16):2 of degree 17 may not be.
All 13 non-abelian simple groups smaller than PSL(2,25) (order 7800) are known to be realizable over $\mathbb{Q}$.
A simple example: cyclic groups.
It is possible, using classical results, to construct explicitly a polynomial whose Galois group over $\mathbb{Q}$ is the cyclic group Z/"n"Z for any positive integer n. To do this, choose a prime p such that "p" ≡ 1 (mod "n"); this is possible by Dirichlet's theorem. Let Q("μ") be the cyclotomic extension of $\mathbb{Q}$ generated by μ, where μ is a primitive "p"-th root of unity; the Galois group of Q("μ")/Q is cyclic of order "p" − 1.
Since n divides "p" − 1, the Galois group has a cyclic subgroup H of order ("p" − 1)/"n". The fundamental theorem of Galois theory implies that the corresponding fixed field, "F" = Q("μ")"H", has Galois group Z/"n"Z over $\mathbb{Q}$. By taking appropriate sums of conjugates of μ, following the construction of Gaussian periods, one can find an element α of F that generates F over $\mathbb{Q}$, and compute its minimal polynomial.
This method can be extended to cover all finite abelian groups, since every such group appears in fact as a quotient of the Galois group of some cyclotomic extension of $\mathbb{Q}$. (This statement should not though be confused with the Kronecker–Weber theorem, which lies significantly deeper.)
Worked example: the cyclic group of order three.
For "n" = 3, we may take "p" = 7. Then Gal(Q("μ")/Q) is cyclic of order six. Let us take the generator η of this group which sends μ to "μ"3. We are interested in the subgroup "H" = {1, "η"3} of order two. Consider the element "α" = "μ" + "η"3("μ"). By construction, α is fixed by H, and only has three conjugates over $\mathbb{Q}$:
"α" = "η"0("α") = "μ" + "μ"6,
"β" = "η"1("α") = "μ"3 + "μ"4,
"γ" = "η"2("α") = "μ"2 + "μ"5.
Using the identity:
1 + "μ" + "μ"2 + ⋯ + "μ"6 = 0,
one finds that
"α" + "β" + "γ" = −1,
"αβ" + "βγ" + "γα" = −2,
"αβγ" = 1.
Therefore α is a root of the polynomial
("x" − "α")("x" − "β")("x" − "γ") = "x"3 + "x"2 − 2"x" − 1,
which consequently has Galois group Z/3Z over $\mathbb{Q}$.
Symmetric and alternating groups.
Hilbert showed that all symmetric and alternating groups are represented as Galois groups of polynomials with rational coefficients.
The polynomial "xn" + "ax" + "b" has discriminant
$(-1)^{\frac{n(n-1)}{2}} \!\left( n^n b^{n-1} + (-1)^{1-n} (n-1)^{n-1} a^n \right)\!.$
We take the special case
"f"("x", "s") = "xn" − "sx" − "s".
Substituting a prime integer for s in "f"("x", "s") gives a polynomial (called a specialization of "f"("x", "s")) that by Eisenstein's criterion is irreducible. Then "f"("x", "s") must be irreducible over $\mathbb{Q}(s)$. Furthermore, "f"("x", "s") can be written
$x^n - \tfrac{x}{2} - \tfrac{1}{2} - \left( s - \tfrac{1}{2} \right)\!(x+1)$
and "f"("x", 1/2) can be factored to:
$\tfrac{1}{2} (x-1)\!\left( 1+ 2x + 2x^2 + \cdots + 2 x^{n-1} \right)$
whose second factor is irreducible (but not by Eisenstein's criterion). Only the reciprocal polynomial is irreducible by Eisenstein's criterion. We have now shown that the group Gal("f"("x", "s")/Q("s")) is doubly transitive.
We can then find that this Galois group has a transposition. Use the scaling (1 − "n")"x" = "ny" to get
$ y^n - \left \{ s \left ( \frac{1-n}{n} \right )^{n-1} \right \} y - \left \{ s \left ( \frac{1-n}{n} \right )^n \right \}$
and with
$ t = \frac{s (1-n)^{n-1}}{n^n},$
we arrive at:
"g"("y", "t") = "yn" − "nty" + ("n" − 1)"t"
which can be arranged to
"yn" − "y" − ("n" − 1)("y" − 1) + ("t" − 1)(−"ny" + "n" − 1).
Then "g"("y", 1) has 1 as a double zero and its other "n" − 2 zeros are simple, and a transposition in Gal("f"("x", "s")/Q("s")) is implied. Any finite doubly transitive permutation group containing a transposition is a full symmetric group.
Hilbert's irreducibility theorem then implies that an infinite set of rational numbers give specializations of "f"("x", "t") whose Galois groups are "Sn" over the rational field $\mathbb{Q}$. In fact this set of rational numbers is dense in $\mathbb{Q}$.
The discriminant of "g"("y", "t") equals
$ (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} (1-t),$
and this is not in general a perfect square.
Alternating groups.
Solutions for alternating groups must be handled differently for odd and even degrees.
Odd Degree.
Let
$t = 1 - (-1)^{\tfrac{n(n-1)}{2}} n u^2$
Under this substitution the discriminant of "g"("y", "t") equals
$\begin{align}
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left (1 - \left (1 - (-1)^{\tfrac{n(n-1)}{2}} n u^2 \right ) \right) \\
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left ((-1)^{\tfrac{n(n-1)}{2}} n u^2 \right ) \\
&= n^{n+1} (n-1)^{n-1} t^{n-1} u^2
\end{align}$
which is a perfect square when n is odd.
Even Degree.
Let:
$t = \frac{1}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2}$
Under this substitution the discriminant of "g"("y", "t") equals:
$\begin{align}
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left (1 - \frac{1}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2} \right ) \\
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left ( \frac{\left ( 1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2 \right ) - 1}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2} \right ) \\
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left ( \frac{(-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2} \right ) \\
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left (t (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2 \right ) \\
&= n^n (n-1)^n t^n u^2
\end{align}$
which is a perfect square when n is even.
Again, Hilbert's irreducibility theorem implies the existence of infinitely many specializations whose Galois groups are alternating groups.
Rigid groups.
Suppose that "C"1, …, "Cn" are conjugacy classes of a finite group G, and A be the set of n-tuples ("g"1, …, "gn") of G such that "gi" is in "Ci" and the product "g"1…"gn" is trivial. Then A is called rigid if it is nonempty, G acts transitively on it by conjugation, and each element of A generates G.
showed that if a finite group G has a rigid set then it can often be realized as a Galois group over a cyclotomic extension of the rationals. (More precisely, over the cyclotomic extension of the rationals generated by the values of the irreducible characters of G on the conjugacy classes "Ci".)
This can be used to show that many finite simple groups, including the monster group, are Galois groups of extensions of the rationals. The monster group is generated by a triad of elements of orders 2, 3, and 29. All such triads are conjugate.
The prototype for rigidity is the symmetric group "Sn", which is generated by an n-cycle and a transposition whose product is an ("n" − 1)-cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group.
A construction with an elliptic modular function.
Let "n" > 1 be any integer. A lattice Λ in the complex plane with period ratio τ has a sublattice Λ′ with period ratio "nτ". The latter lattice is one of a finite set of sublattices permuted by the modular group PSL(2, Z), which is based on changes of basis for Λ. Let j denote the elliptic modular function of Felix Klein. Define the polynomial "φn" as the product of the differences ("X" − "j"(Λ"i")) over the conjugate sublattices. As a polynomial in X, "φn" has coefficients that are polynomials over $\mathbb{Q}$ in "j"("τ").
On the conjugate lattices, the modular group acts as PGL(2, Z/"nZ). It follows that "φn" has Galois group isomorphic to PGL(2, Z/"nZ) over $\mathbb{Q}(\mathrm{J}(\tau))$.
Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing "φn" to polynomials with Galois group PGL(2, Z/"nZ) over $\mathbb{Q}$. The groups PGL(2, Z/"nZ) include infinitely many non-solvable groups.
Notes.
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263714
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abstract_algebra
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Block design in combinatorial mathematics
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a with λ = 1 and "t" = 2 or (recently) "t" ≥ 2.
A Steiner system with parameters "t", "k", "n", written S("t","k","n"), is an "n"-element set "S" together with a set of "k"-element subsets of "S" (called blocks) with the property that each "t"-element subset of "S" is contained in exactly one block. In an alternate notation for block designs, an S("t","k","n") would be a "t"-("n","k",1) design.
This definition is relatively new. The classical definition of Steiner systems also required that "k" = "t" + 1. An S(2,3,"n") was (and still is) called a "Steiner triple" (or "triad") "system", while an S(3,4,"n") is called a "Steiner quadruple system", and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.
Long-standing problems in design theory were whether there exist any nontrivial Steiner systems (nontrivial meaning "t" < "k" < "n") with "t" ≥ 6; also whether infinitely many have "t" = 4 or 5. Both existences were proved by Peter Keevash in 2014. His proof is non-constructive and, as of 2019, no actual Steiner systems are known for large values of "t".
Types of Steiner systems.
A finite projective plane of order "q", with the lines as blocks, is an S(2, "q" + 1, "q"2 + "q" + 1), since it has "q"2 + "q" + 1 points, each line passes through "q" + 1 points, and each pair of distinct points lies on exactly one line.
A finite affine plane of order "q", with the lines as blocks, is an S(2, "q", "q"2). An affine plane of order "q" can be obtained from a projective plane of the same order by removing one block and all of the points in that block from the projective plane. Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.
An S(3,4,"n") is called a Steiner quadruple system. A necessary and sufficient condition for the existence of an S(3,4,"n") is that "n" $\equiv$ 2 or 4 (mod 6). The abbreviation SQS("n") is often used for these systems. Up to isomorphism, SQS(8) and SQS(10) are unique, there are 4 SQS(14)s and 1,054,163 SQS(16)s.
An S(4,5,"n") is called a Steiner quintuple system. A necessary condition for the existence of such a system is that "n" $\equiv$ 3 or 5 (mod 6) which comes from considerations that apply to all the classical Steiner systems. An additional necessary condition is that "n" $\not\equiv$ 4 (mod 5), which comes from the fact that the number of blocks must be an integer. Sufficient conditions are not known. There is a unique Steiner quintuple system of order 11, but none of order 15 or order 17. Systems are known for orders 23, 35, 47, 71, 83, 107, 131, 167 and 243. The smallest order for which the existence is not known (as of 2011) is 21.
Steiner triple systems.
An S(2,3,"n") is called a Steiner triple system, and its blocks are called triples. It is common to see the abbreviation STS("n") for a Steiner triple system of order "n". The total number of pairs is "n(n-1)/2", of which three appear in a triple, and so the total number of triples is "n"("n"−1)/6. This shows that "n" must be of the form "6k+1" or "6k + 3" for some "k". The fact that this condition on "n" is sufficient for the existence of an S(2,3,"n") was proved by Raj Chandra Bose and T. Skolem. The projective plane of order 2 (the Fano plane) is an STS(7) and the affine plane of order 3 is an STS(9). Up to isomorphism, the STS(7) and STS(9) are unique, there are two STS(13)s, 80 STS(15)s, and 11,084,874,829 STS(19)s.
We can define a multiplication on the set "S" using the Steiner triple system by setting "aa" = "a" for all "a" in "S", and "ab" = "c" if {"a","b","c"} is a triple. This makes "S" an idempotent, commutative quasigroup. It has the additional property that "ab" = "c" implies "bc" = "a" and "ca" = "b". Conversely, any (finite) quasigroup with these properties arises from a Steiner triple system. Commutative idempotent quasigroups satisfying this additional property are called "Steiner quasigroups".
Resolvable Steiner systems.
Some of the S(2,3,n) systems can have their triples partitioned into (n-1)/2 sets each having (n/3) pairwise disjoint triples. This is called "resolvable" and such systems are called "Kirkman triple systems" after Thomas Kirkman, who studied such resolvable systems before Steiner. Dale Mesner, Earl Kramer, and others investigated collections of Steiner triple systems that are mutually disjoint (i.e., no two Steiner systems in such a collection share a common triplet). It is known (Bays 1917, Kramer & Mesner 1974) that seven different S(2,3,9) systems can be generated to together cover all 84 triplets on a 9-set; it was also known by them that there are 15360 different ways to find such 7-sets of solutions, which reduce to two non-isomorphic solutions under relabeling, with multiplicities 6720 and 8640 respectively.
The corresponding question for finding thirteen different disjoint S(2,3,15) systems was asked by James Sylvester in 1860 as an extension of the Kirkman's schoolgirl problem, namely whether Kirkman's schoolgirls could march for an entire term of 13 weeks with no triplet of girls being repeated over the whole term. The question was solved by RHF Denniston in 1974, who constructed Week 1 as follows:
for girls labeled A to O, and constructed each subsequent week's solution from its immediate predecessor by changing A to B, B to C, ... L to M and M back to A, all while leaving N and O unchanged. The Week 13 solution, upon undergoing that relabeling, returns to the Week 1 solution. Denniston reported in his paper that the search he employed took 7 hours on an Elliott 4130 computer at the University of Leicester, and he immediately ended the search on finding the solution above, not looking to establish uniqueness. The number of non-isomorphic solutions to Sylvester's problem remains unknown as of 2021.
Properties.
It is clear from the definition of S("t", "k", "n") that $1 < t < k < n$. (Equalities, while technically possible, lead to trivial systems.)
If S("t", "k", "n") exists, then taking all blocks containing a specific element and discarding that element gives a "derived system" S("t"−1, "k"−1, "n"−1). Therefore, the existence of S("t"−1, "k"−1, "n"−1) is a necessary condition for the existence of S("t", "k", "n").
The number of "t"-element subsets in S is $\tbinom n t$, while the number of "t"-element subsets in each block is $\tbinom k t$. Since every "t"-element subset is contained in exactly one block, we have $\tbinom n t = b\tbinom k t$, or
$b = \frac{\tbinom nt}{\tbinom kt} = \frac{n(n-1)\cdots(n-t+1)}{k(k-1)\cdots(k-t+1)},$
where "b" is the number of blocks. Similar reasoning about "t"-element subsets containing a particular element gives us $\tbinom{n-1}{t-1}=r\tbinom{k-1}{t-1}$, or
$r=\frac{\tbinom{n-1}{t-1}}{\tbinom{k-1}{t-1}}$ =$\frac{(n-t+1)\cdots(n-2)(n-1)}{(k-t+1)\cdots(k-2)(k-1)},$
where "r" is the number of blocks containing any given element. From these definitions follows the equation $bk=rn$. It is a necessary condition for the existence of S("t", "k", "n") that "b" and "r" are integers. As with any block design, Fisher's inequality $b\ge n$ is true in Steiner systems.
Given the parameters of a Steiner system S("t, k, n") and a subset of size $t' \leq t$, contained in at least one block, one can compute the number of blocks intersecting that subset in a fixed number of elements by constructing a Pascal triangle. In particular, the number of blocks intersecting a fixed block in any number of elements is independent of the chosen block.
The number of blocks that contain any "i"-element set of points is:
$ \lambda_i = \left.\binom{n-i}{t-i} \right/ \binom{k-i}{t-i} \text{ for } i = 0,1,\ldots,t, $
It can be shown that if there is a Steiner system S(2, "k", "n"), where "k" is a prime power greater than 1, then "n" $\equiv$ 1 or "k" (mod "k"("k"−1)). In particular, a Steiner triple system S(2, 3, "n") must have "n" = 6"m" + 1 or 6"m" + 3. And as we have already mentioned, this is the only restriction on Steiner triple systems, that is, for each natural number "m", systems S(2, 3, 6"m" + 1) and S(2, 3, 6"m" + 3) exist.
History.
Steiner triple systems were defined for the first time by Wesley S. B. Woolhouse in 1844 in the Prize question #1733 of Lady's and Gentlemen's Diary. The posed problem was solved by Thomas Kirkman (1847). In 1850 Kirkman posed a variation of the problem known as Kirkman's schoolgirl problem, which asks for triple systems having an additional property (resolvability). Unaware of Kirkman's work, Jakob Steiner (1853) reintroduced triple systems, and as this work was more widely known, the systems were named in his honor.
Mathieu groups.
Several examples of Steiner systems are closely related to group theory. In particular, the finite simple groups called Mathieu groups arise as automorphism groups of Steiner systems:
The Steiner system S(5, 6, 12).
There is a unique S(5,6,12) Steiner system; its automorphism group is the Mathieu group M12, and in that context it is denoted by W12.
Projective line construction.
This construction is due to Carmichael (1937).
Add a new element, call it ∞, to the 11 elements of the finite field F11 (that is, the integers mod 11). This set, "S", of 12 elements can be formally identified with the points of the projective line over F11. Call the following specific subset of size 6,
$\{\infty,1,3,4,5,9\}, $
a "block" (it contains ∞ together with the 5 nonzero squares in F11). From this block, we obtain the other blocks of the S(5,6,12) system by repeatedly applying the linear fractional transformations:
$z' = f(z) = \frac{az + b}{cz + d},$
where a,b,c,d are in F11 and "ad − bc" = 1.
With the usual conventions of defining "f" (−"d"/"c") = ∞ and "f" (∞) = "a"/"c", these functions map the set "S" onto itself. In geometric language, they are projectivities of the projective line. They form a group under composition which is the projective special linear group PSL(2,11) of order 660. There are exactly five elements of this group that leave the starting block fixed setwise, namely those such that "b=c=0" and "ad"=1 so that "f(z) = a"2 "z". So there will be 660/5 = 132 images of that block. As a consequence of the multiply transitive property of this group acting on this set, any subset of five elements of "S" will appear in exactly one of these 132 images of size six.
Kitten construction.
An alternative construction of W12 is obtained by use of the 'kitten' of R.T. Curtis, which was intended as a "hand calculator" to write down blocks one at a time. The kitten method is based on completing patterns in a 3x3 grid of numbers, which represent an affine geometry on the vector space F3xF3, an S(2,3,9) system.
Construction from K6 graph factorization.
The relations between the graph factors of the complete graph K6 generate an S(5,6,12). A K6 graph has 6 vertices, 15 edges, 15 perfect matchings, and 6 different 1-factorizations (ways to partition the edges into disjoint perfect matchings). The set of vertices (labeled 123456) and the set of factorizations (labeled "ABCDEF") provide one block each. Every pair of factorizations has exactly one perfect matching in common. Suppose factorizations "A" and "B" have the common matching with edges 12, 34 and 56. Add three new blocks "AB"3456, 12"AB"56, and 1234"AB", replacing each edge in the common matching with the factorization labels in turn. Similarly add three more blocks 12"CDEF", 34"CDEF", and 56"CDEF", replacing the factorization labels by the corresponding edge labels of the common matching. Do this for all 15 pairs of factorizations to add 90 new blocks. Finally, take the full set of $\tbinom{12}{6} = 924 $ combinations of 6 objects out of 12, and discard any combination that has 5 or more objects in common with any of the 92 blocks generated so far. Exactly 40 blocks remain, resulting in 2 + 90 + 40 = 132 blocks of the S(5,6,12). This method works because there is an outer automorphism on the symmetric group "S"6, which maps the vertices to factorizations and the edges to partitions. Permuting the vertices causes the factorizations to permute differently, in accordance with the outer automorphism.
The Steiner system S(5, 8, 24).
The Steiner system S(5, 8, 24), also known as the Witt design or Witt geometry, was first described by Carmichael (1931) and rediscovered by Witt (1938). This system is connected with many of the sporadic simple groups and with the exceptional 24-dimensional lattice known as the Leech lattice. The automorphism group of S(5, 8, 24) is the Mathieu group M24, and in that context the design is denoted W24 ("W" for "Witt")
Direct lexicographic generation.
All 8-element subsets of a 24-element set are generated in lexicographic order, and any such subset which differs from some subset already found in fewer than four positions is discarded.
The list of octads for the elements 01, 02, 03, ..., 22, 23, 24 is then:
01 02 03 04 05 06 07 08
01 02 03 04 09 10 11 12
01 02 03 04 13 14 15 16
. (next 753 octads omitted)
13 14 15 16 17 18 19 20
13 14 15 16 21 22 23 24
17 18 19 20 21 22 23 24
Each single element occurs 253 times somewhere in some octad. Each pair occurs 77 times. Each triple occurs 21 times. Each quadruple (tetrad) occurs 5 times. Each quintuple (pentad) occurs once. Not every hexad, heptad or octad occurs.
Construction from the binary Golay code.
The 4096 codewords of the 24-bit binary Golay code are generated, and the 759 codewords with a Hamming weight of 8 correspond to the S(5,8,24) system.
The Golay code can be constructed by many methods, such as generating all 24-bit binary strings in lexicographic order and discarding those that differ from some earlier one in fewer than 8 positions. The result looks like this:
The codewords form a group under the XOR operation.
Projective line construction.
This construction is due to Carmichael (1931).
Add a new element, call it ∞, to the 23 elements of the finite field F23 (that is, the integers mod 23). This set, "S", of 24 elements can be formally identified with the points of the projective line over F23. Call the following specific subset of size 8,
$\{\infty,0,1,3,12,15,21,22\}, $
a "block". (We can take any octad of the extended binary Golay code, seen as a quadratic residue code.) From this block, we obtain the other blocks of the S(5,8,24) system by repeatedly applying the linear fractional transformations:
$z' = f(z) = \frac{az + b}{cz + d},$
where a,b,c,d are in F23 and "ad − bc" = 1.
With the usual conventions of defining "f" (−"d"/"c") = ∞ and "f" (∞) = "a"/"c", these functions map the set "S" onto itself. In geometric language, they are projectivities of the projective line. They form a group under composition which is the projective special linear group PSL(2,23) of order 6072. There are exactly 8 elements of this group that leave the initial block fixed setwise. So there will be 6072/8 = 759 images of that block. These form the octads of S(5,8,24).
Construction from the Miracle Octad Generator.
The Miracle Octad Generator (MOG) is a tool to generate octads, such as those containing specified subsets. It consists of a 4x6 array with certain weights assigned to the rows. In particular, an 8-subset should obey three rules in order to be an octad of S(5,8,24). First, each of the 6 columns should have the same parity, that is, they should all have an odd number of cells or they should all have an even number of cells. Second, the top row should have the same parity as each of the columns. Third, the rows are respectively multiplied by the weights 0, 1, 2, and 3 over the finite field of order 4, and column sums are calculated for the 6 columns, with multiplication and addition using the finite field arithmetic definitions. The resulting column sums should form a valid "hexacodeword" of the form ("a", "b", "c", "a" + "b" + "c", "3a" + "2b" + "c", "2a" + "3b" + "c") where "a, b, c" are also from the finite field of order 4. If the column sums' parities don't match the row sum parity, or each other, or if there do not exist "a, b, c" such that the column sums form a valid hexacodeword, then that subset of 8 is not an octad of S(5,8,24).
The MOG is based on creating a bijection (Conwell 1910, "The three-space PG(3,2) and its group") between the 35 ways to partition an 8-set into two different 4-sets, and the 35 lines of the Fano 3-space PG(3,2). It is also geometrically related (Cullinane, "Symmetry Invariance in a Diamond Ring", Notices of the AMS, pp A193-194, Feb 1979) to the 35 different ways to partition a 4x4 array into 4 different groups of 4 cells each, such that if the 4x4 array represents a four-dimensional finite affine space, then the groups form a set of parallel subspaces.
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In graph theory, the degree diameter problem is the problem of finding the largest possible graph for a given maximum degree and diameter. The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer. The table below gives current progress on this problem (excluding the case of degree 2, where the largest graphs are cycles with an odd number of vertices).
Table of the orders of the largest known graphs for the undirected degree diameter problem.
Below is the table of the vertex numbers for the best-known graphs (as of July 2022) in the undirected degree diameter problem for graphs of degree at most 3 ≤ "d" ≤ 16 and diameter 2 ≤ "k" ≤ 10. Only a few of the graphs in this table (marked in bold) are known to be optimal (that is, largest possible). The remainder are merely the largest so far discovered, and thus finding a larger graph that is closer in order (in terms of the size of the vertex set) to the Moore bound is considered an open problem. Some general constructions are known for values of "d" and "k" outside the range shown in the table.
The following table is the key to the colors in the table presented above:
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In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module "M" is a finite increasing filtration of "M" by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of "M" into its simple constituents.
A composition series may not exist, and when it does, it need not be unique. Nevertheless, a group of results known under the general name Jordan–Hölder theorem asserts that whenever composition series exist, the "isomorphism classes" of simple pieces (although, perhaps, not their "location" in the composition series in question) and their multiplicities are uniquely determined. Composition series may thus be used to define invariants of finite groups and Artinian modules.
A related but distinct concept is a chief series: a composition series is a maximal "subnormal" series, while a chief series is a maximal "normal series".
For groups.
If a group "G" has a normal subgroup "N", then the factor group "G"/"N" may be formed, and some aspects of the study of the structure of "G" may be broken down by studying the "smaller" groups "G/N" and "N". If "G" has no normal subgroup that is different from "G" and from the trivial group, then "G" is a simple group. Otherwise, the question naturally arises as to whether "G" can be reduced to simple "pieces", and if so, are there any unique features of the way this can be done?
More formally, a composition series of a group "G" is a subnormal series of finite length
$1 = H_0\triangleleft H_1\triangleleft \cdots \triangleleft H_n = G,$
with strict inclusions, such that each "H""i" is a maximal proper normal subgroup of "H""i"+1. Equivalently, a composition series is a subnormal series such that each factor group "H""i"+1 / "H""i" is simple. The factor groups are called composition factors.
A subnormal series is a composition series if and only if it is of maximal length. That is, there are no additional subgroups which can be "inserted" into a composition series. The length "n" of the series is called the composition length.
If a composition series exists for a group "G", then any subnormal series of "G" can be "refined" to a composition series, informally, by inserting subgroups into the series up to maximality. Every finite group has a composition series, but not every infinite group has one. For example, $\mathbb{Z}$ has no composition series.
Uniqueness: Jordan–Hölder theorem.
A group may have more than one composition series. However, the Jordan–Hölder theorem (named after Camille Jordan and Otto Hölder) states that any two composition series of a given group are equivalent. That is, they have the same composition length and the same composition factors, up to permutation and isomorphism. This theorem can be proved using the Schreier refinement theorem. The Jordan–Hölder theorem is also true for transfinite "ascending" composition series, but not transfinite "descending" composition series .
gives a short proof of the Jordan–Hölder theorem by intersecting the terms in one subnormal series with those in the other series.
Example.
For a cyclic group of order "n", composition series correspond to ordered prime factorizations of "n", and in fact yields a proof of the fundamental theorem of arithmetic.
For example, the cyclic group $C_{12}$ has $C_1\triangleleft C_2\triangleleft C_6 \triangleleft C_{12}, \ \, C_1\triangleleft C_2\triangleleft C_4\triangleleft C_{12}, $ and $C_1\triangleleft C_3\triangleleft C_6 \triangleleft C_{12}$ as three different composition series. The sequences of composition factors obtained in the respective cases are $C_2,C_3,C_2, \ \, C_2,C_2,C_3, $ and $C_3,C_2,C_2.$
For modules.
The definition of composition series for modules restricts all attention to submodules, ignoring all additive subgroups that are "not" submodules. Given a ring "R" and an "R"-module "M", a composition series for "M" is a series of submodules
$\{0\} = J_0 \subset \cdots \subset J_n = M$
where all inclusions are strict and "J""k" is a maximal submodule of "J""k"+1 for each "k". As for groups, if "M" has a composition series at all, then any finite strictly increasing series of submodules of "M" may be refined to a composition series, and any two composition series for "M" are equivalent. In that case, the (simple) quotient modules "J""k"+1/"J""k" are known as the composition factors of "M," and the Jordan–Hölder theorem holds, ensuring that the number of occurrences of each isomorphism type of simple "R"-module as a composition factor does not depend on the choice of composition series.
It is well known that a module has a finite composition series if and only if it is both an Artinian module and a Noetherian module. If "R" is an Artinian ring, then every finitely generated "R"-module is Artinian and Noetherian, and thus has a finite composition series. In particular, for any field "K", any finite-dimensional module for a finite-dimensional algebra over "K" has a composition series, unique up to equivalence.
Generalization.
Groups with a set of operators generalize group actions and ring actions on a group. A unified approach to both groups and modules can be followed as in or , simplifying some of the exposition. The group "G" is viewed as being acted upon by elements (operators) from a set Ω. Attention is restricted entirely to subgroups invariant under the action of elements from Ω, called Ω-subgroups. Thus Ω-composition series must use only Ω-subgroups, and Ω-composition factors need only be Ω-simple. The standard results above, such as the Jordan–Hölder theorem, are established with nearly identical proofs.
The special cases recovered include when Ω = "G" so that "G" is acting on itself. An important example of this is when elements of "G" act by conjugation, so that the set of operators consists of the inner automorphisms. A composition series under this action is exactly a chief series. Module structures are a case of Ω-actions where Ω is a ring and some additional axioms are satisfied.
For objects in an abelian category.
A composition series of an object "A" in an abelian category is a sequence of subobjects
$A=X_0\supsetneq X_1\supsetneq \dots \supsetneq X_n=0$
such that each quotient object "Xi" /"X""i" + 1 is simple (for 0 ≤ "i" < "n"). If "A" has a composition series, the integer "n" only depends on "A" and is called the length of "A".
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In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead of edges containing 2 vertices in a usual graph).
3-dimensional matching, often abbreviated as 3DM, is also the name of a well-known computational problem: finding a largest 3-dimensional matching in a given hypergraph. 3DM is one of the first problems that were proved to be NP-hard.
Definition.
Let "X", "Y", and "Z" be finite sets, and let "T" be a subset of "X" × "Y" × "Z". That is, "T" consists of triples ("x", "y", "z") such that "x" ∈ "X", "y" ∈ "Y", and "z" ∈ "Z". Now "M" ⊆ "T" is a 3-dimensional matching if the following holds: for any two distinct triples ("x"1, "y"1, "z"1) ∈ "M" and ("x"2, "y"2, "z"2) ∈ "M", we have "x"1 ≠ "x"2, "y"1 ≠ "y"2, and "z"1 ≠ "z"2.
Example.
The figure on the right illustrates 3-dimensional matchings. The set "X" is marked with red dots, "Y" is marked with blue dots, and "Z" is marked with green dots. Figure (a) shows the set "T" (gray areas). Figure (b) shows a 3-dimensional matching "M" with |"M"| = 2, and Figure (c) shows a 3-dimensional matching "M" with |"M"| = 3.
The matching "M" illustrated in Figure (c) is a "maximum 3-dimensional matching", i.e., it maximises |"M"|. The matching illustrated in Figures (b)–(c) are "maximal 3-dimensional matchings", i.e., they cannot be extended by adding more elements from "T".
Here is example interactive visualisation in javascript
Comparison with bipartite matching.
A "2-dimensional matching" can be defined in a completely analogous manner. Let "X" and "Y" be finite sets, and let "T" be a subset of "X" × "Y". Now "M" ⊆ "T" is a 2-dimensional matching if the following holds: for any two distinct pairs ("x"1, "y"1) ∈ "M" and ("x"2, "y"2) ∈ "M", we have "x"1 ≠ "x"2 and "y"1 ≠ "y"2.
In the case of 2-dimensional matching, the set "T" can be interpreted as the set of edges in a bipartite graph "G" = ("X", "Y", "T"); each edge in "T" connects a vertex in "X" to a vertex in "Y". A 2-dimensional matching is then a matching in the graph "G", that is, a set of pairwise non-adjacent edges.
Hence 3-dimensional matchings can be interpreted as a generalization of matchings to hypergraphs: the sets "X", "Y", and "Z" contain the vertices, each element of "T" is a hyperedge, and the set "M" consists of pairwise non-adjacent edges (edges that do not have a common vertex). In case of 2-dimensional matching, we have Y = Z.
Comparison with set packing.
A 3-dimensional matching is a special case of a set packing: we can interpret each element ("x", "y", "z") of "T" as a subset {"x", "y", "z"} of "X" ∪ "Y" ∪ "Z"; then a 3-dimensional matching "M" consists of pairwise disjoint subsets.
Decision problem.
In computational complexity theory, "3-dimensional matching (3DM)" is the name of the following decision problem: given a set "T" and an integer "k", decide whether there exists a 3-dimensional matching "M" ⊆ "T" with |"M"| ≥ "k".
This decision problem is known to be NP-complete; it is one of Karp's 21 NP-complete problems. It is NP-complete even in the special case that "k" = |"X"| = |"Y"| = |"Z"| and when each element is contained in exactly 3 sets, i.e., when we want a perfect matching in a 3-regular hypergraph. In this case, a 3-dimensional matching is not only a set packing, but also an exact cover: the set "M" covers each element of "X", "Y", and "Z" exactly once. The proof is by reduction from 3SAT. Given a 3SAT instance, we construct a 3DM instance as follows:
Special cases.
There exist polynomial time algorithms for solving 3DM in dense hypergraphs.
Optimization problem.
A "maximum 3-dimensional matching" is a largest 3-dimensional matching. In computational complexity theory, this is also the name of the following optimization problem: given a set "T", find a 3-dimensional matching "M" ⊆ "T" that maximizes "|M|".
Since the decision problem described above is NP-complete, this optimization problem is NP-hard, and hence it seems that there is no polynomial-time algorithm for finding a maximum 3-dimensional matching. However, there are efficient polynomial-time algorithms for finding a maximum bipartite matching (maximum 2-dimensional matching), for example, the Hopcroft–Karp algorithm.
Approximation algorithms.
There is a very simple polynomial-time 3-approximation algorithm for 3-dimensional matching: find any maximal 3-dimensional matching. Just like a maximal matching is within factor 2 of a maximum matching, a maximal 3-dimensional matching is within factor 3 of a maximum 3-dimensional matching.
For any constant ε > 0 there is a polynomial-time (4/3 + ε)-approximation algorithm for 3-dimensional matching.
However, attaining better approximation factors is probably hard: the problem is APX-complete, that is, it is hard to approximate within some constant.
It is NP-hard to achieve an approximation factor of 95/94 for maximum 3-d matching, and an approximation factor of 48/47 for maximum 4-d matching. The hardness remains even when restricted to instances with exactly two occurrences of each element.
Parallel algorithms.
There are various algorithms for 3-d matching in the Massively parallel computation model.
Notes.
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The problem of inverting exponentiation in finite groups
In mathematics, for given real numbers "a" and "b", the logarithm log"b" "a" is a number "x" such that "b""x" = "a". Analogously, in any group "G", powers "b""k" can be defined for all integers "k", and the discrete logarithm log"b" "a" is an integer "k" such that "b""k" = "a". In number theory, the more commonly used term is index: we can write "x" = ind"r" "a" (mod "m") (read "the index of "a" to the base "r" modulo "m"") for "r""x" ≡ "a" (mod "m") if "r" is a primitive root of "m" and gcd("a","m") = 1.
Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography, such as ElGamal, base their security on the assumption that the discrete logarithm problem (DLP) over carefully chosen groups has no efficient solution.
Definition.
Let "G" be any group. Denote its group operation by multiplication and its identity element by 1. Let "b" be any element of "G". For any positive integer "k", the expression "b""k" denotes the product of "b" with itself "k" times:
$b^k = \underbrace{b \cdot b \cdots b}_{k \; \text{factors}}.$
Similarly, let "b"−"k" denote the product of "b"−1 with itself "k" times. For "k" = 0, the "k"th power is the identity: "b"0 = 1.
Let "a" also be an element of "G". An integer "k" that solves the equation "b""k" = "a" is termed a discrete logarithm (or simply logarithm, in this context) of "a" to the base "b". One writes "k" = log"b" "a".
Examples.
Powers of 10.
The powers of 10 are
$\ldots, 0.001, 0.01, 0.1, 1, 10, 100, 1000, \ldots.$
For any number "a" in this list, one can compute log10 "a". For example, log10 10000 = 4, and log10 0.001 = −3. These are instances of the discrete logarithm problem.
Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents. For example, the equation log10 53 = 1.724276… means that 101.724276… = 53. While integer exponents can be defined in any group using products and inverses, arbitrary real exponents, such as this 1.724276…, require other concepts such as the exponential function.
In group-theoretic terms, the powers of 10 form a cyclic group "G" under multiplication, and 10 is a generator for this group. The discrete logarithm log10 "a" is defined for any "a" in "G".
Powers of a fixed real number.
A similar example holds for any non-zero real number "b". The powers form a multiplicative subgroup "G" = {…, "b"−3, "b"−2, "b"−1, 1, "b"1, "b"2, "b"3, …} of the non-zero real numbers. For any element "a" of "G", one can compute log"b" "a".
Modular arithmetic.
One of the simplest settings for discrete logarithms is the group (Z"p")×. This is the group of multiplication modulo the prime "p". Its elements are congruence classes modulo "p", and the group product of two elements may be obtained by ordinary integer multiplication of the elements followed by reduction modulo "p".
The "k"th power of one of the numbers in this group may be computed by finding its "k"th power as an integer and then finding the remainder after division by "p". When the numbers involved are large, it is more efficient to reduce modulo "p" multiple times during the computation. Regardless of the specific algorithm used, this operation is called modular exponentiation. For example, consider (Z17)×. To compute 34 in this group, compute 34 = 81, and then divide 81 by 17, obtaining a remainder of 13. Thus 34 = 13 in the group (Z17)×.
The discrete logarithm is just the inverse operation. For example, consider the equation 3"k" ≡ 13 (mod 17) for "k". From the example above, one solution is "k" = 4, but it is not the only solution. Since 316 ≡ 1 (mod 17)—as follows from Fermat's little theorem—it also follows that if "n" is an integer then 34+16"n" ≡ 34 × (316)"n" ≡ 13 × 1"n" ≡ 13 (mod 17). Hence the equation has infinitely many solutions of the form 4 + 16"n". Moreover, because 16 is the smallest positive integer "m" satisfying 3"m" ≡ 1 (mod 17), these are the only solutions. Equivalently, the set of all possible solutions can be expressed by the constraint that "k" ≡ 4 (mod 16).
Powers of the identity.
In the special case where "b" is the identity element 1 of the group "G", the discrete logarithm log"b" "a" is undefined for "a" other than 1, and every integer "k" is a discrete logarithm for "a" = 1.
Properties.
Powers obey the usual algebraic identity "b""k" + "l" = "b""k" "b""l". In other words, the function
$f \colon \mathbf{Z} \to G$
defined by "f"("k") = "b""k" is a group homomorphism from the integers Z under addition onto the subgroup "H" of "G" generated by "b". For all "a" in "H", log"b" "a" exists. Conversely, log"b" "a" does not exist for "a" that are not in "H".
If "H" is infinite, then log"b" "a" is also unique, and the discrete logarithm amounts to a group isomorphism
$\log_b \colon H \to \mathbf{Z}.$
On the other hand, if "H" is finite of order "n", then log"b" "a" is unique only up to congruence modulo "n", and the discrete logarithm amounts to a group isomorphism
$\log_b\colon H \to \mathbf{Z}_n,$
where Z"n" denotes the additive group of integers modulo "n".
The familiar base change formula for ordinary logarithms remains valid: If "c" is another generator of "H", then
$\log_c a = \log_c b \cdot \log_b a.$
Algorithms.
Unsolved problem in computer science:
Can the discrete logarithm be computed in polynomial time on a classical computer?
The discrete logarithm problem is considered to be computationally intractable. That is, no efficient classical algorithm is known for computing discrete logarithms in general.
A general algorithm for computing log"b" "a" in finite groups "G" is to raise "b" to larger and larger powers "k" until the desired "a" is found. This algorithm is sometimes called "trial multiplication". It requires running time linear in the size of the group "G" and thus exponential in the number of digits in the size of the group. Therefore, it is an exponential-time algorithm, practical only for small groups "G".
More sophisticated algorithms exist, usually inspired by similar algorithms for integer factorization. These algorithms run faster than the naïve algorithm, some of them proportional to the square root of the size of the group, and thus exponential in half the number of digits in the size of the group. However none of them runs in polynomial time (in the number of digits in the size of the group).
There is an efficient quantum algorithm due to Peter Shor.
Efficient classical algorithms also exist in certain special cases. For example, in the group of the integers modulo "p" under addition, the power "b""k" becomes a product "bk", and equality means congruence modulo "p" in the integers. The extended Euclidean algorithm finds "k" quickly.
With Diffie–Hellman a cyclic group modulus a prime "p" is used, allowing an efficient computation of the discrete logarithm with Pohlig–Hellman if the order of the group (being "p"−1) is sufficiently smooth, i.e. has no large prime factors.
Comparison with integer factorization.
While computing discrete logarithms and factoring integers are distinct problems, they share some properties:
Cryptography.
There exist groups for which computing discrete logarithms is apparently difficult. In some cases (e.g. large prime order subgroups of groups (Z"p")×) there is not only no efficient algorithm known for the worst case, but the average-case complexity can be shown to be about as hard as the worst case using random self-reducibility.
At the same time, the inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using exponentiation by squaring, for example). This asymmetry is analogous to the one between integer factorization and integer multiplication. Both asymmetries (and other possibly one-way functions) have been exploited in the construction of cryptographic systems.
Popular choices for the group "G" in discrete logarithm cryptography (DLC) are the cyclic groups (Z"p")× (e.g. ElGamal encryption, Diffie–Hellman key exchange, and the Digital Signature Algorithm) and cyclic subgroups of elliptic curves over finite fields ("see" Elliptic curve cryptography).
While there is no publicly known algorithm for solving the discrete logarithm problem in general, the first three steps of the number field sieve algorithm only depend on the group "G", not on the specific elements of "G" whose finite log is desired. By precomputing these three steps for a specific group, one need only carry out the last step, which is much less computationally expensive than the first three, to obtain a specific logarithm in that group.
It turns out that much Internet traffic uses one of a handful of groups that are of order 1024 bits or less, e.g. cyclic groups with order of the Oakley primes specified in RFC 2409. The Logjam attack used this vulnerability to compromise a variety of Internet services that allowed the use of groups whose order was a 512-bit prime number, so called export grade.
The authors of the Logjam attack estimate that the much more difficult precomputation needed to solve the discrete log problem for a 1024-bit prime would be within the budget of a large national intelligence agency such as the U.S. National Security Agency (NSA). The Logjam authors speculate that precomputation against widely reused 1024 DH primes is behind claims in leaked NSA documents that NSA is able to break much of current cryptography.
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abstract_algebra
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Permutation group that preserves no non-trivial partition
In mathematics, a permutation group "G" acting on a non-empty finite set "X" is called primitive if "G" acts transitively on "X" and the only partitions the "G"-action preserves are the trivial partitions into either a single set or into |"X"| singleton sets. Otherwise, if "G" is transitive and "G" does preserve a nontrivial partition, "G" is called imprimitive.
While primitive permutation groups are transitive, not all transitive permutation groups are primitive. The simplest example is the Klein four-group acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the trivial group acting on a 2-element set. This is because for a non-transitive action, either the orbits of "G" form a nontrivial partition preserved by "G", or the group action is trivial, in which case "all" nontrivial partitions of "X" (which exists for |"X"| ≥ 3) are preserved by "G".
This terminology was introduced by Évariste Galois in his last letter, in which he used the French term "équation primitive" for an equation whose Galois group is primitive.
Properties.
In the same letter in which he introduced the term "primitive", Galois stated the following theorem:If "G" is a primitive solvable group acting on a finite set "X", then the order of "X" is a power of a prime number "p". Further, "X" may be identified with an affine space over the finite field with "p" elements, and "G" acts on "X" as a subgroup of the affine group.If the set "X" on which "G" acts is finite, its cardinality is called the "degree" of "G".
A corollary of this result of Galois is that, if p is an odd prime number, then the order of a solvable transitive group of degree p is a divisor of $p(p-1).$ In fact, every transitive group of prime degree is primitive (since the number of elements of a partition fixed by G must be a divisor of p), and $p(p-1)$ is the cardinality of the affine group of an affine space with p elements.
It follows that, if p is a prime number greater than 3, the symmetric group and the alternating group of degree p are not solvable, since their order are greater than $p(p-1).$ Abel–Ruffini theorem results from this and the fact that there are polynomials with a symmetric Galois group.
An equivalent definition of primitivity relies on the fact that every transitive action of a group "G" is isomorphic to an action arising from the canonical action of "G" on the set "G"/"H" of cosets for "H" a subgroup of "G". A group action is primitive if it is isomorphic to "G"/"H" for a "maximal" subgroup "H" of "G", and imprimitive otherwise (that is, if there is a proper subgroup "K" of "G" of which "H" is a proper subgroup). These imprimitive actions are examples of induced representations.
The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937:
There are a large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the symmetric and alternating group, are subgroups of the affine group on the 4-dimensional space over the 2-element finite field.
$\eta=\begin{pmatrix}
Examples.
1 & 2 & 3 \\
2 & 3 & 1 \end{pmatrix}.$
Both $S_3$ and the group generated by $\eta$ are primitive.
$\sigma=\begin{pmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 4 & 1 \end{pmatrix}.$
The group generated by $\sigma$ is not primitive, since the partition $(X_1, X_2)$ where $X_1 = \{1,3\}$ and $X_2 = \{2,4\}$ is preserved under $\sigma$, i.e. $\sigma(X_1) = X_2$ and $\sigma(X_2)=X_1$.
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abstract_algebra
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In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group.
History.
The ping-pong argument goes back to the late 19th century and is commonly attributed to Felix Klein who used it to study subgroups of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space or, equivalently Möbius transformations of the Riemann sphere. The ping-pong lemma was a key tool used by Jacques Tits in his 1972 paper containing the proof of a famous result now known as the Tits alternative. The result states that a finitely generated linear group is either virtually solvable or contains a free subgroup of rank two. The ping-pong lemma and its variations are widely used in geometric topology and geometric group theory.
Modern versions of the ping-pong lemma can be found in many books such as Lyndon & Schupp, de la Harpe, Bridson & Haefliger and others.
Formal statements.
Ping-pong lemma for several subgroups.
This version of the ping-pong lemma ensures that several subgroups of a group acting on a set generate a free product. The following statement appears in Olijnyk and Suchchansky (2004), and the proof is from de la Harpe (2000).
Let "G" be a group acting on a set "X" and let "H"1, "H"2, ..., "H""k" be subgroups of "G" where "k" ≥ 2, such that at least one of these subgroups has order greater than 2.
Suppose there exist pairwise disjoint nonempty subsets "X"1, "X"2, ...,"X""k" of "X" such that the following holds:
Then <math display="block">\langle H_1,\dots, H_k\rangle=H_1\ast\dots \ast H_k.$
Proof.
By the definition of free product, it suffices to check that a given (nonempty) reduced word represents a nontrivial element of $G$. Let $w$ be such a word of length $m\geq 2$, and let <math display="block">w = \prod_{i=1}^m w_i,$ where <math display="inline">w_i \in H_{\alpha_i}$ for some <math display="inline">\alpha_i \in \{1,\dots,k\}$. Since <math display="inline">w$ is reduced, we have $\alpha_i \neq \alpha_{i+1}$ for any $i = 1, \dots, m-1$ and each $w_i$ is distinct from the identity element of $H_{\alpha_i}$. We then let $w$ act on an element of one of the sets <math display="inline">X_i$. As we assume that at least one subgroup $H_i$ has order at least 3, without loss of generality we may assume that $H_1$ has order at least 3. We first make the assumption that $\alpha_1$and $\alpha_m$ are both 1 (which implies $m \geq 3$). From here we consider $w$ acting on $X_2$. We get the following chain of containments:
<math display="block">w(X_2) \subseteq \prod_{i=1}^{m-1} w_i(X_1) \subseteq \prod_{i=1}^{m-2} w_i(X_{\alpha_{m-1}}) \subseteq \dots \subseteq w_1(X_{\alpha_2}) \subseteq X_1.$
By the assumption that different $X_i$'s are disjoint, we conclude that $w$ acts nontrivially on some element of $X_2$, thus $w$ represents a nontrivial element of $G$.
To finish the proof we must consider the three cases:
In each case, $hwh^{-1}$ after reduction becomes a reduced word with its first and last letter in $H_1$. Finally, $hwh^{-1}$ represents a nontrivial element of $G$, and so does $w$. This proves the claim.
The Ping-pong lemma for cyclic subgroups.
Let "G" be a group acting on a set "X". Let "a"1, ...,"a""k" be elements of "G" of infinite order, where "k" ≥ 2. Suppose there exist disjoint nonempty subsets
"X"1+, ..., "X""k"+ and "X"1–, ..., "X""k"–
of "X" with the following properties:
Then the subgroup "H" = ⟨"a"1, ..., "a""k"⟩ ≤ "G" generated by "a"1, ..., "a""k" is free with free basis {"a"1, ..., "a""k"}.
Proof.
This statement follows as a corollary of the version for general subgroups if we let "X""i" = "X""i"+ ∪ "X""i"− and let "H""i" = ⟨"a""i"⟩.
Examples.
Special linear group example.
One can use the ping-pong lemma to prove that the subgroup "H" = ⟨"A","B"⟩ ≤ SL2(Z), generated by the matrices
<math display="block">A = \begin{pmatrix}1 & 2\\ 0 &1 \end{pmatrix}$ and <math display="block">B = \begin{pmatrix}1 & 0\\ 2 &1 \end{pmatrix}$
is free of rank two.
Proof.
Indeed, let "H"1 = ⟨"A"⟩ and "H"2 = ⟨"B"⟩ be cyclic subgroups of SL2(Z) generated by "A" and "B" accordingly. It is not hard to check that "A" and "B" are elements of infinite order in SL2(Z) and that
<math display="block">H_1 = \{A^n \mid n\in \Z\} = \left\{\begin{pmatrix}1 & 2n\\ 0 & 1 \end{pmatrix} : n\in\Z\right\}$
and
<math display="block">H_2 = \{B^n \mid n\in \Z\} = \left\{\begin{pmatrix}1 & 0\\ 2n & 1 \end{pmatrix} : n\in\Z\right\}.$
Consider the standard action of SL2(Z) on R2 by linear transformations. Put
<math display="block">X_1 = \left\{ \begin{pmatrix}x \\ y \end{pmatrix}\in \R^2 : |x|>|y|\right\}$
and
<math display="block">X_2 = \left\{ \begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb R^2 : |x|<|y|\right\}.$
It is not hard to check, using the above explicit descriptions of "H"1 and "H"2, that for every nontrivial "g" ∈ "H"1 we have "g"("X"2) ⊆ "X"1 and that for every nontrivial "g" ∈ "H"2 we have "g"("X"1) ⊆ "X"2. Using the alternative form of the ping-pong lemma, for two subgroups, given above, we conclude that "H" = "H"1 ∗ "H"2. Since the groups "H"1 and "H"2 are infinite cyclic, it follows that "H" is a free group of rank two.
Word-hyperbolic group example.
Let "G" be a word-hyperbolic group which is torsion-free, that is, with no nonidentity elements of finite order. Let "g", "h" ∈ "G" be two non-commuting elements, that is such that "gh" ≠ "hg". Then there exists "M" ≥ 1 such that for any integers "n" ≥ "M", "m" ≥ "M" the subgroup H = ⟨"g""n", "h""m"⟩ ≤ "G" is free of rank two.
Sketch of the proof.
The group "G" acts on its "hyperbolic boundary" ∂"G" by homeomorphisms. It is known that if "a" in "G" is a nonidentity element then "a" has exactly two distinct fixed points, "a"∞ and "a"−∞ in ∂"G" and that "a"∞ is an attracting fixed point while "a"−∞ is a repelling fixed point.
Since "g" and "h" do not commute, basic facts about word-hyperbolic groups imply that "g"∞, "g"−∞, "h"∞ and "h"−∞ are four distinct points in ∂"G". Take disjoint neighborhoods "U"+, "U"–, "V"+, and "V"– of "g"∞, "g"−∞, "h"∞ and "h"−∞ in ∂"G" respectively.
Then the attracting/repelling properties of the fixed points of "g" and "h" imply that there exists "M" ≥ 1 such that for any integers "n" ≥ "M", "m" ≥ "M" we have:
The ping-pong lemma now implies that "H" = ⟨"g""n", "h""m"⟩ ≤ "G" is free of rank two.
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2274857
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abstract_algebra
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Fundamental result in the branch of mathematics known as character theory
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, within representation theory of a finite group.
Background.
A precursor to Brauer's induction theorem was Artin's induction theorem, which states that |"G"| times the trivial character of "G" is an integer combination of characters which are each induced from trivial characters of cyclic subgroups of "G." Brauer's theorem removes the factor |"G"|,
but at the expense of expanding the collection of subgroups used. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such induction theorem (with integer combinations of characters induced from linear characters) could be proved with a collection of subgroups smaller than the Brauer elementary subgroups.
Another result between Artin's induction theorem and Brauer's induction theorem, also due to Brauer and also known as "Brauer's theorem" or "Brauer's lemma" is the fact that the regular representation of "G" can be written as $1+\sum\lambda_i\rho_i$ where the $\lambda_i$ are "positive rationals" and the $\rho_i$ are induced from characters of cyclic subgroups of "G". Note that in Artin's theorem the characters are induced from the trivial character of the cyclic group, while here they are induced from arbitrary characters (in applications to Artin's "L" functions it is important that the groups are cyclic and hence all characters are linear giving that the corresponding "L" functions are analytic).
Statement.
Let "G" be a finite group and let Char("G") denote the subring of the ring of complex-valued class functions of "G" consisting of integer combinations of irreducible characters. Char("G") is known as the character ring of "G", and its elements are known as virtual characters (alternatively, as generalized characters, or sometimes difference characters). It is a ring by virtue of the fact that the product of characters of "G" is again a character of "G." Its multiplication is given by the elementwise product of class functions.
Brauer's induction theorem shows that the character ring can be generated (as an abelian group) by induced characters of the form $\lambda^{G}_{H}$, where "H" ranges over subgroups of "G" and λ ranges over linear characters (having degree 1) of "H".
In fact, Brauer showed that the subgroups "H" could be chosen from a very
restricted collection, now called Brauer elementary subgroups. These are direct products of cyclic groups and groups whose order is a power of a prime.
Proofs.
The proof of Brauer's induction theorem exploits the ring structure of Char("G") (most proofs also make use of a slightly larger ring, Char*(G), which consists of $\mathbb{Z}[\omega]$-combinations of irreducible characters, where ω is a primitive complex |"G"|-th root of unity). The set of integer combinations of characters induced from linear characters of Brauer elementary subgroups is an ideal "I"("G") of Char("G"), so the proof reduces to showing that the trivial character is in "I"("G"). Several proofs of the theorem, beginning with a proof due to Brauer and John Tate, show that the trivial character is in the analogously defined ideal "I"*("G") of Char*("G") by concentrating attention on one prime "p" at a time, and constructing integer-valued elements of "I"*("G") which differ (elementwise) from the trivial character by (integer multiples of) a sufficiently high power of "p." Once this is achieved for every prime divisor of |"G"|, some manipulations with congruences
and algebraic integers, again exploiting the fact that "I"*("G") is an ideal of Ch*("G"), place the trivial character in "I"("G"). An auxiliary result here is that a $\mathbb{Z}[\omega]$-valued class function lies in the ideal "I"*("G") if its values are all divisible (in $\mathbb{Z}[\omega]$) by |"G"|.
Brauer's induction theorem was proved in 1946, and there are now many alternative proofs. In 1986, Victor Snaith gave a proof by a radically different approach, topological in nature (an application of the Lefschetz fixed-point theorem). There has been related recent work on the question of finding natural and explicit forms of Brauer's theorem, notably by Robert Boltje.
Applications.
Using Frobenius reciprocity, Brauer's induction theorem leads easily to his fundamental characterization of characters, which asserts that a complex-valued class function of "G" is a virtual character if and only if its restriction to each Brauer elementary subgroup of "G" is a virtual character. This result, together with the fact that a virtual character θ is an irreducible character
if and only if θ(1) "> 0" and $\langle \theta,\theta \rangle =1 $ (where $\langle,\rangle$ is the usual inner product on the ring of complex-valued class functions) gives
a means of constructing irreducible characters without explicitly constructing the associated representations.
An initial motivation for Brauer's induction theorem was application to Artin L-functions. It shows that those are built up from Dirichlet L-functions, or more general Hecke L-functions. Highly significant for that application is whether each character of "G" is a "non-negative" integer combination of characters induced from linear characters of subgroups. In general, this is not the case. In fact, by a theorem of Taketa, if all characters of "G" are so expressible, then "G" must be a solvable group (although solvability alone does not guarantee such expressions- for example, the solvable group "SL(2,3)" has an irreducible complex character of degree 2 which is not expressible as a non-negative integer combination of characters induced from linear characters of subgroups). An ingredient of the proof of Brauer's induction theorem is that when "G" is a finite nilpotent group, every complex irreducible character of "G" is induced from a linear character of some subgroup.
Notes.
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abstract_algebra
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Non-commutative group with 6 elements
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group.
This page illustrates many group concepts using this group as example.
Symmetry groups.
The dihedral group D3 is the symmetry group of an equilateral triangle, that is, it is the set of all transformations such as reflection, rotation, and combinations of these, that leave the shape and position of this triangle fixed. In the case of D3, every possible permutation of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is isomorphic to the symmetric group S3 of all permutations of three distinct elements. This is not the case for dihedral groups of higher orders.
The dihedral group D3 is isomorphic to two other symmetry groups in three dimensions:
Permutations of a set of three objects.
Consider three colored blocks (red, green, and blue), initially placed in the order RGB. The symmetric group S3 is then the group of all possible rearrangements of these blocks.
If we denote by "a" the action "swap the first two blocks", and by "b" the action "swap the last two blocks", we can write all possible permutations in terms of these two actions.
In multiplicative form, we traditionally write "xy" for the combined action "first do "y", then do "x""; so that "ab" is the action RGB ↦ RBG ↦ BRG, i.e., "take the last block and move it to the front".
If we write "e" for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:
The notation in brackets is the cycle notation.
Note that the action "aa" has the effect RGB ↦ GRB ↦ RGB, leaving the blocks as they were; so we can write "aa" = "e".
Similarly,
so each of the above actions has an inverse.
By inspection, we can also determine associativity and closure (two of the necessary group axioms); note for example that
The group is non-abelian since, for example, "ab" ≠ "ba". Since it is built up from the basic actions "a" and "b", we say that the set {"a", "b"} "generates" it.
The group has presentation
$\langle r, a \mid r^3 = 1, a^2 = 1, ara = r^{-1} \rangle$, also written $\langle r, a \mid r^3, a^2, arar \rangle$
or
$\langle a, b \mid a^2 = b^2 = (ab)^3 = 1 \rangle$, also written $\langle a, b \mid a^2, b^2, (ab)^3 \rangle$
where "a" and "b" are swaps and "r" = "ab" is a cyclic permutation. Note that the second presentation means that the group is a Coxeter group. (In fact, all dihedral and symmetry groups are Coxeter groups.)
Summary of group operations.
With the generators "a" and "b", we define the additional shorthands "c" := "aba", "d" := "ab" and "f" := "ba", so that "a, b, c, d, e", and "f" are all the elements of this group. We can then summarize the group operations in the form of a Cayley table:
Note that non-equal non-identity elements only commute if they are each other's inverse. Therefore, the group is centerless, i.e., the center of the group consists only of the identity element.
Conjugacy classes.
We can easily distinguish three kinds of permutations of the three blocks, the conjugacy classes of the group:
For example, (RG) and (RB) are both of the form ("x" "y"); a permutation of the letters R, G, and B (namely (GB)) changes the notation (RG) into (RB). Therefore, if we apply (GB), then (RB), and then the inverse of (GB), which is also (GB), the resulting permutation is (RG).
Note that conjugate group elements always have the same order, but in general two group elements that have the same order need not be conjugate.
Subgroups.
From Lagrange's theorem we know that any non-trivial subgroup of a group with 6 elements must have order 2 or 3. In fact the two cyclic permutations of all three blocks, with the identity, form a subgroup of order 3, index 2, and the swaps of two blocks, each with the identity, form three subgroups of order 2, index 3. The existence of subgroups of order 2 and 3 is also a consequence of Cauchy's theorem.
The first-mentioned is { (), (RGB), (RBG) }, the alternating group A3.
The left cosets and the right cosets of A3 coincide (as they do for any subgroup of index 2) and consist of A3 and the set of three swaps { (RB), (RG), (BG)}.
The left cosets of { (), (RG) } are:
The right cosets of { (RG), () } are:
Thus A3 is normal, and the other three non-trivial subgroups are not. The quotient group "G" / "A"3 is isomorphic with "C"2.
$G = \mathrm{A}_3 \rtimes H$, a semidirect product, where "H" is a subgroup of two elements: () and one of the three swaps. This decomposition is also a consequence (particular case) of the Schur–Zassenhaus theorem.
In terms of permutations the two group elements of "G" / A3 are the set of even permutations and the set of odd permutations.
If the original group is that generated by a 120°-rotation of a plane about a point, and reflection with respect to a line through that point, then the quotient group has the two elements which can be described as the subsets "just rotate (or do nothing)" and "take a mirror image".
Note that for the symmetry group of a "square", an uneven permutation of vertices does "not" correspond to taking a mirror image, but to operations not allowed for "rectangles", i.e. 90° rotation and applying a diagonal axis of reflection.
Semidirect products.
$\mathrm{C}_3 \rtimes_\varphi \mathrm{C}_2$ is $\mathrm{C}_3 \times \mathrm{C}_2$ if both "φ"(0) and "φ"(1) are the identity.
The semidirect product is isomorphic to the dihedral group of order 6 if "φ"(0) is the identity and "φ"(1) is the non-trivial automorphism of C3, which inverses the elements.
Thus we get:
("n"1, 0) * ("n"2, "h"2) = ("n"1 + "n"2, "h"2)
("n"1, 1) * ("n"2, "h"2) = ("n"1 − "n"2, 1 + "h"2)
for all "n"1, "n"2 in C3 and "h"2 in C2.
More concisely,
$(n_1, h_1) * (n_2, h_2) = (n_1 + (-1)^{h_1} n_2, h_1 + h_2)$
for all "n"1, "n"2 in C3 and "h"1, "h"2 in C2.
In a Cayley table:
Note that for the second digit we essentially have a 2×2 table, with 3×3 equal values for each of these 4 cells. For the first digit the left half of the table is the same as the right half, but the top half is different from the bottom half.
For the "direct" product the table is the same except that the first digits of the bottom half of the table are the same as in the top half.
Group action.
Consider "D"3 in the geometrical way, as a symmetry group of isometries of the plane, and consider the corresponding group action on a set of 30 evenly spaced points on a circle, numbered 0 to 29, with 0 at one of the reflexion axes.
This section illustrates group action concepts for this case.
The action of "G" on "X" is called
Orbits and stabilizers.
The orbit of a point "x" in "X" is the set of elements of "X" to which "x" can be moved by the elements of "G". The orbit of "x" is denoted by "Gx":
$Gx = \left\{ g\cdot x \mid g \in G \right\}$
The orbits are {0, 10, 20}, {1, 9, 11, 19, 21, 29}, {2, 8, 12, 18, 22, 28}, {3, 7, 13, 17, 23, 27}, {4, 6, 14, 16, 24, 26}, and {5, 15, 25}. The points within an orbit are "equivalent". If a symmetry group applies for a pattern, then within each orbit the color is the same.
The set of all orbits of "X" under the action of "G" is written as "X" / "G".
If "Y" is a subset of "X", we write "GY" for the set { "g" · "y" : "y" ∈ "Y" and "g" ∈ "G" }. We call the subset "Y" "invariant under G" if "GY" = "Y" (which is equivalent to "GY" ⊆ "Y"). In that case, "G" also operates on "Y". The subset "Y" is called "fixed under G" if "g" · "y" = "y" for all "g" in "G" and all "y" in "Y". The union of e.g. two orbits is invariant under "G", but not fixed.
For every "x" in "X", we define the stabilizer subgroup of "x" (also called the isotropy group or little group) as the set of all elements in "G" that fix "x":
$G_x = \{g \in G \mid g\cdot x = x\}$
If "x" is a reflection point (0, 5, 10, 15, 20, or 25), its stabilizer is the group of order two containing the identity and the reflection in "x". In other cases the stabilizer is the trivial group.
For a fixed "x" in "X", consider the map from "G" to "X" given by "g" ↦ "g" · "x". The image of this map is the orbit of "x" and the coimage is the set of all left cosets of "Gx". The standard quotient theorem of set theory then gives a natural bijection between "G" / "G""x" and "Gx". Specifically, the bijection is given by "hGx" ↦ "h" · "x". This result is known as the orbit-stabilizer theorem. In the two cases of a small orbit, the stabilizer is non-trivial.
If two elements "x" and "y" belong to the same orbit, then their stabilizer subgroups, "G""x" and "G""y", are isomorphic. More precisely: if "y" = "g" · "x", then "G""y" = "gG""x" "g"−1. In the example this applies e.g. for 5 and 25, both reflection points. Reflection about 25 corresponds to a rotation of 10, reflection about 5, and rotation of −10.
A result closely related to the orbit-stabilizer theorem is Burnside's lemma:
$\left|X/G\right|=\frac{1}{\left|G\right|}\sum_{g\in G}\left|X^g\right|$
where "X""g" is the set of points fixed by "g". I.e., the number of orbits is equal to the average number of points fixed per group element.
For the identity all 30 points are fixed, for the two rotations none, and for the three reflections two each: {0, 15}, {5, 20}, and {10, 25}. Thus, the average is six, the number of orbits.
Representation theory.
Up to isomorphism, this group has three irreducible complex unitary representations, which we will call $I$ (the trivial representation), $\rho_1$ and $\rho_2$, where the subscript indicates the dimension. By its definition as a permutation group over the set with three elements, the group has a representation on $\mathbb{C}^3$ by permuting the entries of the vector, the fundamental representation. This representation is not irreducible, as it decomposes as a direct sum of $I$ and $\rho_2$. $I$ appears as the subspace of vectors of the form $(\lambda, \lambda, \lambda), \lambda \in \mathbb{C}$ and $\rho_2$ is the representation on its orthogonal complement, which are vectors of the form $(\lambda_1, \lambda_2, -\lambda_1 -\lambda_2)$.
The nontrivial one-dimensional representation $\rho_1$ arises through the groups $\mathbb{Z}_2$ grading: The action is multiplication by the sign of the permutation of the group element. Every finite group has such a representation since it is a subgroup of a cyclic group by its regular action. Counting the square dimensions of the representations ($1^2 + 1^2 + 2^2 = 6$, the order of the group), we see these must be all of the irreducible representations.
A 2-dimensional irreducible linear representation yields a 1-dimensional projective representation (i.e., an action on the projective line, an embedding in the Möbius group PGL(2, C)), as elliptic transforms. This can be represented by matrices with entries 0 and ±1 (here written as fractional linear transformations), known as the anharmonic group:
and thus descends to a representation over any field, which is always faithful/injective (since no two terms differ only by only a sign). Over the field with two elements, the projective line has only 3 points, and this is thus the exceptional isomorphism $S_3 \approx \mathrm{PGL}(2, 2).$ In characteristic 3, this embedding stabilizes the point $-1 = [-1:1],$ since $2 = 1/2 = -1$ (in characteristic greater than 3 these points are distinct and permuted, and are the orbit of the harmonic cross-ratio). Over the field with three elements, the projective line has 4 elements, and since PGL(2, 3) is isomorphic to the symmetric group on 4 elements, S4, the resulting embedding $\mathrm{S}_3 \hookrightarrow \mathrm{S}_4$ equals the stabilizer of the point $-1$.
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The following list in mathematics contains the finite groups of small order up to group isomorphism.
Counts.
For "n" = 1, 2, … the number of nonisomorphic groups of order "n" is
1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... (sequence in the OEIS)
For labeled groups, see OEIS: .
Glossary.
Each group is named by Small Groups library as G"o""i", where "o" is the order of the group, and "i" is the index used to label the group within that order.
Common group names:
The notations Z"n" and Dih"n" have the advantage that point groups in three dimensions C"n" and D"n" do not have the same notation. There are more isometry groups than these two, of the same abstract group type.
The notation "G" × "H" denotes the direct product of the two groups; "G""n" denotes the direct product of a group with itself "n" times. "G" ⋊ "H" denotes a semidirect product where "H" acts on "G"; this may also depend on the choice of action of "H" on "G".
Abelian and simple groups are noted. (For groups of order "n" < 60, the simple groups are precisely the cyclic groups Z"n", for prime "n".) The equality sign ("=") denotes isomorphism.
The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.
In the lists of subgroups, the trivial group and the group itself are not listed. Where there are several isomorphic subgroups, the number of such subgroups is indicated in parentheses.
Angle brackets <relations> show the presentation of a group.
List of small abelian groups.
The finite abelian groups are either cyclic groups, or direct products thereof; see Abelian group. The numbers of nonisomorphic abelian groups of orders "n" = 1, 2, ... are
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, ... (sequence in the OEIS)
For labeled abelian groups, see OEIS: .
List of small non-abelian groups.
The numbers of non-abelian groups, by order, are counted by (sequence in the OEIS). However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are
6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, ... (sequence in the OEIS)
Classifying groups of small order.
Small groups of prime power order "p""n" are given as follows:
Most groups of small order have a Sylow "p" subgroup "P" with a normal "p"-complement "N" for some prime "p" dividing the order, so can be classified in terms of the possible primes "p", "p"-groups "P", groups "N", and actions of "P" on "N". In some sense this reduces the classification of these groups to the classification of "p"-groups. Some of the small groups that do not have a normal "p"-complement include:
The smallest order for which it is "not" known how many nonisomorphic groups there are is 2048 = 211.
Small Groups Library.
The GAP computer algebra system contains a package called the "Small Groups library," which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups:
It contains explicit descriptions of the available groups in computer readable format.
The smallest order for which the Small Groups library does not have information is 1024.
Notes.
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Zero divisors in a module
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element.
This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements.
This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is allowed by the fact that the abelian groups are the modules over the ring of integers (in fact, this is the origin of the terminology, that has been introduced for abelian groups before being generalized to modules).
In the case of groups that are noncommutative, a "torsion element" is an element of finite order. Contrary to the commutative case, the torsion elements do not form a subgroup, in general.
Definition.
An element "m" of a module "M" over a ring "R" is called a "torsion element" of the module if there exists a regular element "r" of the ring (an element that is neither a left nor a right zero divisor) that annihilates "m", i.e., "r" "m" = 0.
In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element, but this definition does not work well over more general rings.
A module "M" over a ring "R" is called a "torsion module" if all its elements are torsion elements, and "torsion-free" if zero is the only torsion element. If the ring "R" is commutative then the set of all torsion elements forms a submodule of "M", called the "torsion submodule" of "M", sometimes denoted T("M"). If "R" is not commutative, T("M") may or may not be a submodule. It is shown in that "R" is a right Ore ring if and only if T("M") is a submodule of "M" for all right "R"-modules. Since right Noetherian domains are Ore, this covers the case when "R" is a right Noetherian domain (which might not be commutative).
More generally, let "M" be a module over a ring "R" and "S" be a multiplicatively closed subset of "R". An element "m" of "M" is called an "S"-torsion element if there exists an element "s" in "S" such that "s" annihilates "m", i.e., "s" "m" = 0. In particular, one can take for "S" the set of regular elements of the ring "R" and recover the definition above.
An element "g" of a group "G" is called a "torsion element" of the group if it has finite order, i.e., if there is a positive integer "m" such that "g""m" = "e", where "e" denotes the identity element of the group, and "g""m" denotes the product of "m" copies of "g". A group is called a "torsion (or periodic) group" if all its elements are torsion elements, and a torsion-free group if its only torsion element is the identity element. Any abelian group may be viewed as a module over the ring Z of integers, and in this case the two notions of torsion coincide.
Case of a principal ideal domain.
Suppose that "R" is a (commutative) principal ideal domain and "M" is a finitely generated "R"-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module "M" up to isomorphism. In particular, it claims that
$M \simeq F\oplus T(M),$
where "F" is a free "R"-module of finite rank (depending only on "M") and T("M") is the torsion submodule of "M". As a corollary, any finitely generated torsion-free module over "R" is free. This corollary "does not" hold for more general commutative domains, even for "R" = K["x","y"], the ring of polynomials in two variables.
For non-finitely generated modules, the above direct decomposition is not true. The torsion subgroup of an abelian group may not be a direct summand of it.
Torsion and localization.
Assume that "R" is a commutative domain and "M" is an "R"-module. Let "Q" be the quotient field of the ring "R". Then one can consider the "Q"-module
$M_Q = M \otimes_R Q,$
obtained from "M" by extension of scalars. Since "Q" is a field, a module over "Q" is a vector space, possibly infinite-dimensional. There is a canonical homomorphism of abelian groups from "M" to "M""Q", and the kernel of this homomorphism is precisely the torsion submodule T("M"). More generally, if "S" is a multiplicatively closed subset of the ring "R", then we may consider localization of the "R"-module "M",
$M_S = M \otimes_R R_S,$
which is a module over the localization "R""S". There is a canonical map from "M" to "M""S", whose kernel is precisely the "S"-torsion submodule of "M".
Thus the torsion submodule of "M" can be interpreted as the set of the elements that "vanish in the localization". The same interpretation continues to hold in the non-commutative setting for rings satisfying the Ore condition, or more generally for any right denominator set "S" and right "R"-module "M".
Torsion in homological algebra.
The concept of torsion plays an important role in homological algebra. If "M" and "N" are two modules over a commutative domain "R" (for example, two abelian groups, when "R" = Z), Tor functors yield a family of "R"-modules Tor"i" ("M","N"). The "S"-torsion of an "R"-module "M" is canonically isomorphic to Tor"R"1("M", "R""S"/"R") by the exact sequence of Tor"R"*: The short exact sequence $0\to R\to R_S \to R_S/R \to 0$ of "R"-modules yields an exact sequence $0\to\operatorname{Tor}^R_1(M, R_S/R)\to M\to M_S$, hence $\operatorname{Tor}^R_1(M, R_S/R)$ is the kernel of the localisation map of "M". The symbol Tor denoting the functors reflects this relation with the algebraic torsion. This same result holds for non-commutative rings as well as long as the set "S" is a right denominator set.
Abelian varieties.
The torsion elements of an abelian variety are "torsion points" or, in an older terminology, "division points". On elliptic curves they may be computed in terms of division polynomials.
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Mathematical group with trivial abelianization
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that "G"(1) = "G" (the commutator subgroup equals the group), or equivalently one such that "G"ab = {1} (its abelianization is trivial).
Examples.
The smallest (non-trivial) perfect group is the alternating group "A"5. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. Conversely, a perfect group need not be simple; for example, the special linear group over the field with 5 elements, SL(2,5) (or the binary icosahedral group, which is isomorphic to it) is perfect but not simple (it has a non-trivial center containing $\left(\begin{smallmatrix}-1 & 0 \\ 0 & -1\end{smallmatrix}\right) = \left(\begin{smallmatrix}4 & 0 \\ 0 & 4\end{smallmatrix}\right)$).
The direct product of any two simple non-abelian groups is perfect but not simple; the commutator of two elements is [("a","b"),("c","d")] = (["a","c"],["b","d"]). Since commutators in each simple group form a generating set, pairs of commutators form a generating set of the direct product.
More generally, a quasisimple group (a perfect central extension of a simple group) that is a non-trivial extension (and therefore not a simple group itself) is perfect but not simple; this includes all the insoluble non-simple finite special linear groups SL("n","q") as extensions of the projective special linear group PSL("n","q") (SL(2,5) is an extension of PSL(2,5), which is isomorphic to "A"5). Similarly, the special linear group over the real and complex numbers is perfect, but the general linear group GL is never perfect (except when trivial or over $\mathbb{F}_2$, where it equals the special linear group), as the determinant gives a non-trivial abelianization and indeed the commutator subgroup is SL.
A non-trivial perfect group, however, is necessarily not solvable; and 4 divides its order (if finite), moreover, if 8 does not divide the order, then 3 does.
Every acyclic group is perfect, but the converse is not true: "A"5 is perfect but not acyclic (in fact, not even superperfect), see . In fact, for $n\ge 5$ the alternating group $A_n$ is perfect but not superperfect, with $H_2(A_n,\Z) = \Z/2$ for $n \ge 8$.
Any quotient of a perfect group is perfect. A non-trivial finite perfect group that is not simple must then be an extension of at least one smaller simple non-abelian group. But it can be the extension of more than one simple group. In fact, the direct product of perfect groups is also perfect.
Every perfect group "G" determines another perfect group "E" (its universal central extension) together with a surjection "f": "E" → "G" whose kernel is in the center of "E,"
such that "f" is universal with this property. The kernel of "f" is called the Schur multiplier of "G" because it was first studied by Issai Schur in 1904; it is isomorphic to the homology group $H_2(G)$.
In the plus construction of algebraic K-theory, if we consider the group $\operatorname{GL}(A) = \text{colim} \operatorname{GL}_n(A)$ for a commutative ring $A$, then the subgroup of elementary matrices $E(R)$ forms a perfect subgroup.
Ore's conjecture.
As the commutator subgroup is "generated" by commutators, a perfect group may contain elements that are products of commutators but not themselves commutators. Øystein Ore proved in 1951 that the alternating groups on five or more elements contained only commutators, and conjectured that this was so for all the finite non-abelian simple groups. Ore's conjecture was finally proven in 2008. The proof relies on the classification theorem.
Grün's lemma.
A basic fact about perfect groups is Grün's lemma from : the quotient of a perfect group by its center is centerless (has trivial center).
Proof: If "G" is a perfect group, let "Z"1 and "Z"2 denote the first two terms of the upper central series of "G" (i.e., "Z"1 is the center of "G", and "Z"2/"Z"1 is the center of "G"/"Z"1). If "H" and "K" are subgroups of "G", denote the commutator of "H" and "K" by ["H", "K"] and note that ["Z"1, "G"] = 1 and ["Z"2, "G"] ⊆ "Z"1, and consequently (the convention that ["X", "Y", "Z"] = [["X", "Y"], "Z"] is followed):
$[Z_2,G,G]=[[Z_2,G],G]\subseteq [Z_1,G]=1$
$[G,Z_2,G]=[[G,Z_2],G]=[[Z_2,G],G]\subseteq [Z_1,G]=1.$
By the three subgroups lemma (or equivalently, by the Commutator#Identities (group theory)|Hall-Witt identity), it follows that ["G", "Z"2] = "G", "G"], "Z"2] = ["G", "G", "Z"2] = {1}. Therefore, "Z"2 ⊆ "Z"1 = "Z"("G"), and the center of the quotient group "G" / "Z"("G") is the [[trivial group.
As a consequence, all Center (group theory)#Higher centers|higher centers (that is, higher terms in the upper central series) of a perfect group equal the center.
Group homology.
In terms of group homology, a perfect group is precisely one whose first homology group vanishes: "H"1("G", Z) = 0, as the first homology group of a group is exactly the abelianization of the group, and perfect means trivial abelianization. An advantage of this definition is that it admits strengthening:
Quasi-perfect group.
Especially in the field of algebraic K-theory, a group is said to be quasi-perfect if its commutator subgroup is perfect; in symbols, a quasi-perfect group is one such that "G"(1) = "G"(2) (the commutator of the commutator subgroup is the commutator subgroup), while a perfect group is one such that "G"(1) = "G" (the commutator subgroup is the whole group). See and .
Notes.
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Math theorem in the field of representation theory
Itô's theorem is a result in the mathematical discipline of representation theory due to Noboru Itô. It generalizes the well-known result that the dimension of an irreducible representation of a group must divide the order of that group.
Statement.
Given an irreducible representation V of a finite group G and a maximal normal abelian subgroup "A" ⊆ "G", the dimension of V must divide ["G":"A"].
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In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of "n" numbers. It is an instance of a scheme for numbering permutations and is an example of an inversion table.
The Lehmer code is named in reference to Derrick Henry Lehmer, but the code had been known since 1888 at least.
The code.
The Lehmer code makes use of the fact that there are
$n!=n\times(n-1)\times\cdots\times2\times1$
permutations of a sequence of "n" numbers. If a permutation "σ" is specified by the sequence ("σ"1, …, "σ""n") of its images of 1, …, "n", then it is encoded by a sequence of "n" numbers, but not all such sequences are valid since every number must be used only once. By contrast the encodings considered here choose the first number from a set of "n" values, the next number from a fixed set of "n" − 1 values, and so forth decreasing the number of possibilities until the last number for which only a single fixed value is allowed; "every" sequence of numbers chosen from these sets encodes a single permutation. While several encodings can be defined, the Lehmer code has several additional useful properties; it is the sequence
$L(\sigma)=(L(\sigma)_1,\ldots,L(\sigma)_n)\quad\text{where}\quad L(\sigma)_i=\#\{ j>i : \sigma_j<\sigma_i \},$
in other words the term "L"("σ")"i" counts the number of terms in ("σ"1, …, "σ""n") to the right of "σ""i" that are smaller than it, a number between 0 and "n" − "i", allowing for "n" + 1 − "i" different values.
A pair of indices ("i","j") with "i" < "j" and "σ""i" > "σ""j" is called an inversion of "σ", and "L"("σ")"i" counts the number of inversions ("i","j") with "i" fixed and varying "j". It follows that "L"("σ")1 + "L"("σ")2 + … + "L"("σ")"n" is the total number of inversions of "σ", which is also the number of adjacent transpositions that are needed to transform the permutation into the identity permutation. Other properties of the Lehmer code include that the lexicographical order of the encodings of two permutations is the same as that of their sequences ("σ"1, …, "σ""n"), that any value 0 in the code represents a right-to-left minimum in the permutation (i.e., a "σ""i" smaller than any "σ""j" to its right), and a value "n" − "i"
at position "i" similarly signifies a right-to-left maximum, and that the Lehmer code of "σ" coincides with the factorial number system representation of its position in the list of permutations of "n" in lexicographical order (numbering the positions starting from 0).
Variations of this encoding can be obtained by counting inversions ("i","j") for fixed "j" rather than fixed "i", by counting inversions with a fixed smaller "value" "σ""j" rather than smaller index "i", or by counting non-inversions rather than inversions; while this does not produce a fundamentally different type of encoding, some properties of the encoding will change correspondingly. In particular counting inversions with a fixed smaller value "σ""j" gives the inversion table of "σ", which can be seen to be the Lehmer code of the inverse permutation.
Encoding and decoding.
The usual way to prove that there are "n"! different permutations of "n" objects is to observe that the first object can be chosen in "n" different ways, the next object in "n" − 1 different ways (because choosing the same number as the first is forbidden), the next in "n" − 2 different ways (because there are now 2 forbidden values), and so forth. Translating this freedom of choice at each step into a number, one obtains an encoding algorithm, one that finds the Lehmer code of a given permutation. One need not suppose the objects permuted to be numbers, but one needs a total ordering of the set of objects. Since the code numbers are to start from 0, the appropriate number to encode each object "σ""i" by is the number of objects that were available at that point (so they do not occur before position "i"), but which are smaller than the object "σ""i" actually chosen. (Inevitably such objects must appear at some position "j" > "i", and ("i","j") will be an inversion, which shows that this number is indeed "L"("σ")"i".)
This number to encode each object can be found by direct counting, in several ways (directly counting inversions, or correcting the total number of objects smaller than a given one, which is its sequence number starting from 0 in the set, by those that are unavailable at its position). Another method which is in-place, but not really more efficient, is to start with the permutation of {0, 1, … "n" − 1} obtained by representing each object by its mentioned sequence number, and then for each entry "x", in order from left to right, correct the items to its right by subtracting 1 from all entries (still) greater than "x" (to reflect the fact that the object corresponding to "x" is no longer available). Concretely a Lehmer code for the permutation B,F,A,G,D,E,C of letters, ordered alphabetically, would first give the list of sequence numbers 1,5,0,6,3,4,2, which is successively transformed
$ \begin{matrix}
\mathbf1&5&0&6&3&4&2\\
1&\mathbf4&0&5&2&3&1\\
1&4&\mathbf0&4&2&3&1\\
1&4&0&\mathbf3&1&2&0\\
1&4&0&3&\mathbf1&2&0\\
1&4&0&3&1&\mathbf1&0\\
1&4&0&3&1&1&\mathbf0\\
$
where the final line is the Lehmer code (at each line one subtracts 1 from the larger entries to the right of the boldface element to form the next line).
For decoding a Lehmer code into a permutation of a given set, the latter procedure may be reversed: for each entry "x", in order from right to left, correct the items to its right by adding 1 to all those (currently) greater than or equal to "x"; finally interpret the resulting permutation of {0, 1, … "n" − 1} as sequence numbers (which amounts to adding 1 to each entry if a permutation of {1, 2, … "n"} is sought). Alternatively the entries of the Lehmer code can be processed from left to right, and interpreted as a number determining the next choice of an element as indicated above; this requires maintaining a list of available elements, from which each chosen element is removed. In the example this would mean choosing element 1 from {A,B,C,D,E,F,G} (which is B) then element 4 from {A,C,D,E,F,G} (which is F), then element 0 from {A,C,D,E,G} (giving A) and so on, reconstructing the sequence B,F,A,G,D,E,C.
Applications to combinatorics and probabilities.
Independence of relative ranks.
The Lehmer code defines a bijection from the symmetric group "S""n" to the Cartesian product $[n]\times[n-1]\times\cdots\times[2]\times[1]$, where ["k"] designates the "k"-element set $\{0,1,\ldots,k-1\}$. As a consequence, under the uniform distribution on "S""n", the component "L"("σ")"i" defines a uniformly distributed random variable on ["n" − "i"], and these random variables are mutually independent, because they are projections on different factors of a Cartesian product.
Number of right-to-left minima and maxima.
Definition : In a sequence "u
(uk)1≤k≤n", there is right-to-left minimum (resp. maximum) at rank "k" if "uk" is strictly smaller (resp. strictly bigger) than each element "ui" with "i>k", i.e., to its right.
Let "B(k)" (resp. "H(k)") be the event "there is right-to-left minimum (resp. maximum) at rank "k"", i.e. "B(k)" is the set of the permutations $\scriptstyle\ \mathfrak{S}_n$ which exhibit a right-to-left minimum (resp. maximum) at rank "k". We clearly have
$\{\omega\in B(k)\}\Leftrightarrow\{L(k,\omega)=0\}\quad\text{and}\quad\{\omega\in H(k)\}\Leftrightarrow\{L(k,\omega)=k-1\}.$
Thus the number "Nb(ω)" (resp. "Nh(ω)") of right-to-left minimum (resp. maximum) for the permutation "ω" can be written as a sum of independent Bernoulli random variables each with a respective parameter of 1/k :
$N_b(\omega)=\sum_{1\le k\le n}\ 1\!\!1_{B(k)}\quad\text{and}\quad N_b(\omega)=\sum_{1\le k\le n}\ 1\!\!1_{H(k)}.$
Indeed, as "L(k)" follows the uniform law on $\scriptstyle\ [\![1,k]\!],$
$\mathbb{P}(B(k))=\mathbb{P}(L(k)=0)=\mathbb{P}(H(k))=\mathbb{P}(L(k)=k-1)=\tfrac1k.$
The generating function for the Bernoulli random variable $1\!\!1_{B(k)}$ is
$G_k(s)=\frac{k-1+s}k,$
therefore the generating function of "Nb" is
$G(s)=\prod_{k=1}^nG_k(s)\ =\ \frac{s^{\overline{n}}}{n!}$
(using the rising factorial notation),
which allows us to recover the product formula for the generating function of the
Stirling numbers of the first kind (unsigned).
The secretary problem.
This is an optimal stop problem, a classic in decision theory, statistics and applied probabilities, where a random permutation is gradually revealed through the first elements of its Lehmer code, and where the goal is to stop exactly at the element k such as σ(k)=n, whereas the only available information (the k first values of the Lehmer code) is not sufficient to compute σ(k).
In less mathematical words : a series of n applicants are interviewed one after the other. The interviewer must hire the best applicant, but must make his decision (“Hire” or “Not hire”) on the spot, without interviewing the next applicant (and "a fortiori" without interviewing all applicants).
The interviewer thus knows the rank of the kth applicant, therefore, at the moment of making his kth decision, the interviewer knows only the k first elements of the Lehmer code whereas he would need to know all of them to make a well informed decision.
To determine the optimal strategies (i.e. the strategy maximizing the probability of a win), the statistical properties of the Lehmer code are crucial.
Allegedly, Johannes Kepler clearly exposed this secretary problem to a friend of his at a time when he was trying to make up his mind and choose one out eleven prospective brides as his second wife. His first marriage had been an unhappy one, having been arranged without himself being consulted, and he was thus very concerned that he could reach the right decision.
Similar concepts.
Two similar vectors are in use. One of them is often called inversion vector, e.g. by Wolfram Alpha.
See also .
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Mathematical theorem
In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
Definitions.
We say that a group "G" satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups of "G":
$1 = G_0 \le G_1 \le G_2 \le \cdots\, $
is eventually constant, i.e., there exists "N" such that "G""N" = "G""N"+1 = "G""N"+2 = ... . We say that "G" satisfies the ACC on normal subgroups if every such sequence of normal subgroups of "G" eventually becomes constant.
Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups:
$G = G_0 \ge G_1 \ge G_2 \ge \cdots.\, $
Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group $\mathbf{Z}$ satisfies ACC but not DCC, since (2) > (2)2 > (2)3 > ... is an infinite decreasing sequence of subgroups. On the other hand, the $p^\infty$-torsion part of $\mathbf{Q}/\mathbf{Z}$ (the quasicyclic "p"-group) satisfies DCC but not ACC.
We say a group "G" is indecomposable if it cannot be written as a direct product of non-trivial subgroups "G" = "H" × "K".
Statement.
If $G$ is a group that satisfies either ACC or DCC on normal subgroups, then there is exactly one way of writing $G$ as a direct product $G_1 \times G_2 \times\cdots \times G_k\,$ of finitely many indecomposable subgroups of $G$. Here, uniqueness means direct decompositions into indecomposable subgroups have the exchange property. That is: suppose $G = H_1 \times H_2 \times \cdots \times H_l\,$ is another expression of $G$ as a product of indecomposable subgroups. Then $k=l$ and there is a reindexing of the $H_i$'s satisfying
Proof.
Proving existence is relatively straightforward: let S be the set of all normal subgroups that can not be written as a product of indecomposable subgroups. Moreover, any indecomposable subgroup is (trivially) the one-term direct product of itself, hence decomposable. If Krull-Schmidt fails, then S contains G; so we may iteratively construct a descending series of direct factors; this contradicts the DCC. One can then invert the construction to show that "all" direct factors of G appear in this way.
The proof of uniqueness, on the other hand, is quite long and requires a sequence of technical lemmas. For a complete exposition, see.
Remark.
The theorem does not assert the existence of a non-trivial decomposition, but merely that any such two decompositions (if they exist) are the same.
Remak decomposition.
A Remak decomposition, introduced by Robert Remak, is a decomposition of an abelian group or similar object into a finite direct sum of indecomposable objects. The Krull–Schmidt theorem gives conditions for a Remak decomposition to exist and for its factors to be unique.
Krull–Schmidt theorem for modules.
If $E \neq 0$ is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian or – equivalently – of finite length), then $E$ is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism.
In general, the theorem fails if one only assumes that the module is Noetherian or Artinian.
History.
The present-day Krull–Schmidt theorem was first proved by Joseph Wedderburn ("Ann. of Math" (1909)), for finite groups, though he mentions some credit is due to an earlier study of G.A. Miller where direct products of abelian groups were considered. Wedderburn's theorem is stated as an exchange property between direct decompositions of maximum length. However, Wedderburn's proof makes no use of automorphisms.
The thesis of Robert Remak (1911) derived the same uniqueness result as Wedderburn but also proved (in modern terminology) that the group of central automorphisms acts transitively on the set of direct decompositions of maximum length of a finite group. From that stronger theorem Remak also proved various corollaries including that groups with a trivial center and perfect groups have a unique Remak decomposition.
Otto Schmidt ("Sur les produits directs, S. M. F. Bull. 41" (1913), 161–164), simplified the main theorems of Remak to the 3 page predecessor to today's textbook proofs. His method improves Remak's use of idempotents to create the appropriate central automorphisms. Both Remak and Schmidt published subsequent proofs and corollaries to their theorems.
Wolfgang Krull ("Über verallgemeinerte endliche Abelsche Gruppen, M. Z. 23" (1925) 161–196), returned to G.A. Miller's original problem of direct products of abelian groups by extending to abelian operator groups with ascending and descending chain conditions. This is most often stated in the language of modules. His proof observes that the idempotents used in the proofs of Remak and Schmidt can be restricted to module homomorphisms; the remaining details of the proof are largely unchanged.
O. Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebras by proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chain conditions. This proof makes no use of idempotents and does not reprove the transitivity of Remak's theorems.
Kurosh's "The Theory of Groups" and Zassenhaus' "The Theory of Groups" include the proofs of Schmidt and Ore under the name of Remak–Schmidt but acknowledge Wedderburn and Ore. Later texts use the title Krull–Schmidt (Hungerford's Algebra) and Krull–Schmidt–Azumaya (Curtis–Reiner). The name Krull–Schmidt is now popularly substituted for any theorem concerning uniqueness of direct products of maximum size. Some authors choose to call direct decompositions of maximum-size Remak decompositions to honor his contributions.
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