Symmetric group

Algebraic structure → Group theory Group theory
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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.^[1] 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) ${\displaystyle G}$.