Symmetric Group

The symmetric group on a finite set is the group formed by every possible bijection, or permutation, of that set to itself, with the group operation being the composition of two permutations one after another. The symmetric group on n elements, written S sub n, has exactly n factorial elements, one for every possible ordering of the set, and it is abelian, meaning the order of composition does not matter, only when n is 2 or smaller. Cayley's theorem shows that every group whatsoever is isomorphic to some subgroup of a symmetric group, making the symmetric groups a kind of universal building block for all of group theory. The symmetric group is also central to Galois theory, invariant theory, the representation theory of Lie groups and combinatorics.