symmetric group

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 $\mathrm_n$ 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 $\mathrm_n$ 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 = \$.

The symmetric group on a set $X$ is denoted in various ways, including $\mathrm_X$, $\mathfrak_X$, $\Sigma_X$, $X!$, and $\operatorname(X)$. If $X$ is the set $\$ then the name may be abbreviated to $\mathrm_n$, $\mathfrak_n$, $\Sigma_n$, or $\operatorname(n)$.

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 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):

: $f = (1\ 3)(4\ 5)=\begin 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end$
: $g = (1\ 2\ 5)(3\ 4)=\begin 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end.$

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
: $fg = f\circ g = (1\ 2\ 4)(3\ 5)=\begin 1 & 2 &3 & 4 & 5 \\ 2 & 4 & 5 & 1 & 3\end.$

A cycle of length , taken to the ''k''th power, will decompose into ''k'' cycles of length ''m'': For example, (, ),
: $(1~2~3~4~5~6)^2 = (1~3~5) (2~4~6).$

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.
# The operation of function composition is closed in the set of permutations of the given set ''X''.
# Function composition is always associative.
# The trivial bijection that assigns each element of ''X'' to itself serves as an identity for the group.
# Every bijection has an inverse function that undoes its action, and thus each element of a symmetric group does have an inverse which is a permutation too.

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:

:$\operatornamef = \begin +1, & \textf\mbox \\ -1, & \textf \text. \end$

With this defini