Definition

See Symmetric group

An important special case is a cycle. All disjoint cycles commute (disjoint refers to the fact that they cycle different elements). Theorem: All elements in the symmetric group can be written as a product of disjoint cycles. If the union of the supports of the cycles equals the $\{1...n\}$, the decomposition is unique and we call it the Cycle decomposition

Proof The support of the cycles in the cycle decompositoin of permutation $\sigma$ are the Group orbits generated by $\sigma$ acting on the elements of $\{1...n\}$ (where the group action is just by interpreting $\sigma$ as a bijection). Corollary: every element in symmetric group can be written as a product of Transpositions, because any cycle is a product of transpositions.

The Conjugacy class of an element in symmetric group is determined by the length of the cycles in the cycle decomposition. So there is a 1-1 correspondence between conjugacy classes and paritions of the set [n].

Lemma. Two permutations are conjugated if and only if they have the same type.

Corollary. The number of conjugacy classes of S_n is equal to the number of partitions of [n], which is $2^n$.

Sign of a permutation