Symmetric group

The symmetric group on a finite set ${\displaystyle X}$ is the group whose elements are all bijective functions from ${\displaystyle X}$ to ${\displaystyle X}$ and whose group operation is that of function composition.[1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree ${\displaystyle n}$ is the symmetric group on the set ${\displaystyle X=\{1,2,\ldots ,n\}}$.

The symmetric group on a set ${\displaystyle X}$ is denoted in various ways, including ${\displaystyle \mathrm {S} _{X}}$, ${\displaystyle {\mathfrak {S}}_{X}}$, ${\displaystyle \Sigma _{X}}$, ${\displaystyle X!}$, and ${\displaystyle \operatorname {Sym} (X)}$.[1] If ${\displaystyle X}$ is the set ${\displaystyle \{1,2,\ldots ,n\}}$ then the name may be abbreviated to ${\displaystyle \mathrm {S} _{n}}$, ${\displaystyle {\mathfrak {S}}_{n}}$, ${\displaystyle \Sigma _{n}}$, or ${\displaystyle \operatorname {Sym} (n)}$.[1]

Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in (Scott 1987, Ch. 11), (Dixon & Mortimer 1996, Ch. 8), and (Cameron 1999).

The symmetric group on a set of ${\displaystyle n}$ elements has order ${\displaystyle n!}$ (the factorial of ${\displaystyle n}$).[2] It is abelian if and only if ${\displaystyle n}$ is less than or equal to 2.[3] For ${\displaystyle n=0}$ and ${\displaystyle n=1}$ (the empty set and the singleton set), the symmetric groups are trivial (they have order ${\displaystyle 0!=1!=1}$). The group S_n is solvable if and only if ${\displaystyle n\leq 4}$. This is an essential part of the proof of the Abel–Ruffini theorem that shows that for every ${\displaystyle n>4}$ there are polynomials of degree ${\displaystyle n}$ 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.

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.

The elements of the symmetric group on a set X are the permutations of X.

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,

$${\displaystyle k={\begin{pmatrix}1&2&3&4&5\\1&4&3&2&5\end{pmatrix}},}$$

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,

$${\displaystyle (2~4)\circ (1~2~3)(4~5)\circ (2~4)=(1~4~3)(2~5).}$$

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 = (μ₁, μ₂, ..., μ_k) with ${\textstyle n=\sum _{i=1}^{k}\mu _{i}}$ and μ₁ ≥ μ₂ ≥ ... ≥ μ_k, is associated the set C_μ of permutations with cycles of lengths μ₁, μ₂, ..., μ_k. Then C_μ is a conjugacy class of S_n, whose elements are said to be of cycle-type ${\displaystyle \mu }$.

The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.

S₀ and S₁

The symmetric groups on the empty set and the singleton set are trivial, which corresponds to 0! = 1! = 1. In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S₀, its only member is the empty function.

S₂

This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group and is thus abelian. In Galois theory, this corresponds to the fact that the quadratic formula gives a direct solution to the general quadratic polynomial after extracting only a single root. In invariant theory, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting f_s(x, y) = f(x, y) + f(y, x), and f_a(x, y) = f(x, y) − f(y, x), one gets that 2⋅f = f_s + f_a. This process is known as symmetrization.

S₃

S₃ is the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from S₃ to S₂ corresponds to the resolving quadratic for a cubic polynomial, as discovered by Gerolamo Cardano, while the A₃ kernel corresponds to the use of the discrete Fourier transform of order 3 in the solution, in the form of Lagrange resolvents.

S₄

The group S₄ is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, 9, 8 and 6 permutations, of the cube.[5] Beyond the group A₄, S₄ has a Klein four-group V as a proper normal subgroup, namely the even transpositions {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, with quotient S₃. In Galois theory, this map corresponds to the resolving cubic to a quartic polynomial, which allows the quartic to be solved by radicals, as established by Lodovico Ferrari. The Klein group can be understood in terms of the Lagrange resolvents of the quartic. The map from S₄ to S₃ also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree n of dimension below n − 1, which only occurs for n = 4.

S₅

S₅ is the first non-solvable symmetric group. Along with the special linear group SL(2, 5) and the icosahedral group A₅ × S₂, S₅ is one of the three non-solvable groups of order 120, up to isomorphism. S₅ is the Galois group of the general quintic equation, and the fact that S₅ is not a solvable group translates into the non-existence of a general formula to solve quintic polynomials by radicals. There is an exotic inclusion map S₅ → S₆ as a transitive subgroup; the obvious inclusion map S_n → S_n+1 fixes a point and thus is not transitive. This yields the outer automorphism of S₆, discussed below, and corresponds to the resolvent sextic of a quintic.

S₆

Unlike all other symmetric groups, S₆, has an outer automorphism. Using the language of Galois theory, this can also be understood in terms of Lagrange resolvents. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map S₅ → S₆ as a transitive subgroup (the obvious inclusion map S_n → S_n+1 fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S₆—see Automorphisms of the symmetric and alternating groups for details.

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.

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.

The symmetric group on n letters is generated by the adjacent transpositions ${\displaystyle \sigma _{i}=(i,i+1)}$ that swap i and i + 1.[6] The collection ${\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}$ generates S_n subject to the following relations:[7]

• ${\displaystyle \sigma _{i}^{2}=1,}$
• ${\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}$ for ${\displaystyle |i-j|>1}$, and
• ${\displaystyle (\sigma _{i}\sigma _{i+1})^{3}=1,}$

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.[8]

A subgroup of a symmetric group is called a permutation group.

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.

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 (Schreier & Ulam 1936) according to (Dixon & Mortimer 1996, p. 259).

The group homology of S_n is quite regular and stabilizes: the first homology (concretely, the abelianization) is:

${\displaystyle H_{1}(\mathrm {S} _{n},\mathbf {Z} )={\begin{cases}0&n<2\\\mathbf {Z} /2&n\geq 2.\end{cases}}}$

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:

${\displaystyle H_{2}(\mathrm {S} _{n},\mathbf {Z} )={\begin{cases}0&n<4\\\mathbf {Z} /2&n\geq 4.\end{cases}}}$

This was computed in (Schur 1911), and corresponds to the double cover of the symmetric group, 2 · S_n.

Note that the exceptional low-dimensional homology of the alternating group (${\displaystyle H_{1}(\mathrm {A} _{3})\cong H_{1}(\mathrm {A} _{4})\cong \mathrm {C} _{3},}$ corresponding to non-trivial abelianization, and ${\displaystyle H_{2}(\mathrm {A} _{6})\cong H_{2}(\mathrm {A} _{7})\cong \mathrm {C} _{6},}$ 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 ${\displaystyle \mathrm {A} _{4}\twoheadrightarrow \mathrm {C} _{3}}$ extends to ${\displaystyle \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3},}$ and the triple covers of A₆ and A₇ extend to triple covers of S₆ and S₇ – but these are not homological – the map ${\displaystyle \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3}}$ 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 (Nakaoka 1961), with the cohomology algebra forming a Hopf algebra.

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 KS_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.

• Braid group
• History of group theory
• Signed symmetric group and Generalized symmetric group
• Symmetry in quantum mechanics § Exchange symmetry or permutation symmetry
• Symmetric inverse semigroup
• Symmetric power

• "Symmetric group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Symmetric group". MathWorld.
• Weisstein, Eric W. "Symmetric group graph". MathWorld.
• Marcus du Sautoy: Symmetry, reality's riddle (video of a talk)
• OEIS Entries dealing with the Symmetric Group