What is "symmetric group"

n. (context mathematics English) A group whose elements are precisely all of the bijections of some set with itself and whose operation is composition of those bijections.

In abstract algebra, the symmetric group S on a finite set of n symbols is the group whose elements are all the permutation operations that can be performed on n distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself. Since there are n! (n factorial) possible permutation operations that can be performed on a tuple composed of n symbols, it follows that the order (the number of elements) of the symmetric group S is n!.

Although symmetric groups can be defined on infinite sets as well, this article discusses only 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 G.