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 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 An 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.
The elements of the symmetric group on a set X are the permutations of X.
The group operation in a symmetric group is function composition, denoted by the symbol ∘ or simply by juxtaposition 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):
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
A cycle of length L = k · m, taken to the kth power, will decompose into k cycles of length m: For example, (k = 2, m = 3),
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:
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.
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 move 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 Sn 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.
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.
Note that the reverse on n elements and perfect shuffle on 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic.
The conjugacy classes of Sn correspond to the cycle structures of permutations; that is, two elements of Sn are conjugate in Sn 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 Sn 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:
This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is,
The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.
Other than the trivial map Sn → C1 ≅ S0 ≅ S1 and the sign map Sn → S2, the most notable homomorphisms between symmetric groups, in order of relative dimension, are:
For n ≥ 5, the alternating group An is simple, and the induced quotient is the sign map: An → Sn → S2 which is split by taking a transposition of two elements. Thus Sn is the semidirect product An ⋊ S2, and has no other proper normal subgroups, as they would intersect An in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in An (and thus themselves be An or Sn).
Sn acts on its subgroup An by conjugation, and for n ≠ 6, Sn is the full automorphism group of An: Aut(An) ≅ Sn. Conjugation by even elements are inner automorphisms of An while the outer automorphism of An of order 2 corresponds to conjugation by an odd element. For n = 6, there is an exceptional outer automorphism of An so Sn is not the full automorphism group of An.
For n ≥ 5, Sn is an almost simple group, as it lies between the simple group An and its group of automorphisms.
Sn can be embedded into An+2 by appending the transposition (n + 1, n + 2) to all odd permutations, while embedding into An+1 is impossible for n > 1.
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.
The normal subgroups of the finite symmetric groups are well understood. If n ≤ 2, Sn 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 Sn, 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 (Scott 1987, Ch. 11.3) or (Dixon & Mortimer 1996, Ch. 8.1).
The maximal subgroups of the finite symmetric groups fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the form Sym(k) × Sym(n − k) for 1 ≤ k < n/2. The imprimitive maximal subgroups are exactly those of the form Sym(k) wr Sym(n/k) where 2 ≤ k ≤ n/2 is a proper divisor of n and "wr" denotes the wreath product acting imprimitively. 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, (Liebeck, Praeger & Saxl 1988) gave a fairly satisfactory description of the maximal subgroups of this type according to (Dixon & Mortimer 1996, p. 268).
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 Fp(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 pn are sometimes denoted Wp(n), and using this notation one has that Wp(n + 1) is the wreath product of Wp(n) and Wp(1).
In general, the Sylow p-subgroups of the symmetric group of degree n are a direct product of ai copies of Wp(i), where 0 ≤ ai ≤ p − 1 and n = a0 + p⋅a1 + ... + pk⋅ak (the base p expansion of n).
These calculations are attributed to (Kaloujnine 1948) and described in more detail in (Rotman 1995, p. 176). Note however that (Kerber 1971, p. 26) attributes the result to an 1844 work of Cauchy, and mentions that it is even covered in textbook form in (Netto 1882, §39–40).
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.
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.
The homology "stabilizes" in the sense of stable homotopy theory: there is an inclusion map Sn → Sn+1, and for fixed k, the induced map on homology Hk(Sn) → Hk(Sn+1) is an isomorphism for sufficiently high n. This is analogous to the homology of families Lie groups stabilizing.
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 Sn 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 KSn 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.