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 ${\displaystyle \mathrm {S} _{n}}$ defined over a finite set of ${\displaystyle n}$ symbols consists of the permutations that can be performed on the ${\displaystyle n}$ symbols. Since there are ${\displaystyle n!}$ (${\displaystyle n}$ factorial) such permutation operations, the order (number of elements) of the symmetric group ${\displaystyle \mathrm {S} _{n}}$ is ${\displaystyle n!}$.

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 ${\displaystyle G}$ is isomorphic to a subgroup of the symmetric group on (the underlying set of) ${\displaystyle G}$.

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. 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 ${\displaystyle \operatorname {Sym} (X)}$. 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 ${\displaystyle \operatorname {Sym} (n)}$.

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 ${\displaystyle n}$). It is abelian if and only if ${\displaystyle n}$ is less than or equal to 2. 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.