Symmetric group

In mathematics, the symmetric group on n symbols, S_n, is the group composed by all permutations of n symbols.

It has n! elements (as many as the permutations of n elements).

Its [conjugacy classes]? correspond to the cycle structures of permutations; that is, two elements of S_n are conjugate if and only if they consist of the same number of cycles of the same lengths. For instance, in S₅, (123)(45) and (143)(25) are conjugate; (123)(45) and (12)(3)(45) are not.