Symmetric group

In abstract algebra, the symmetric group S_n 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_n 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.

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 = { 1, 2, ..., n }.

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