Permutation cycle

A permutation cycle $(a_1{a}_2\dots a_k)$ sends $a_i$ to $a_{i+1}$, sends $a_k$ to $a_1$, and fixes every other element. Its order is $k$.

The cycle type of a permutation is the multiset of lengths in its decomposition into disjoint permutation cycles, including one-cycles for fixed points. Two permutations in the same symmetric group are conjugate exactly when they have the same cycle type.

A three-cycle is a permutation cycle of length three. On any chosen set of three letters, the two possible three-cycles are inverses of one another.

Permutation cycles with disjoint supports commute. The order of their product is the least common multiple of their lengths.