Cycle of a permutation

A cycle $(a_1\ a_2\ \cdots\ a_r)$ is the permutation sending $a_j$ to $a_{j+1}$ , $a_r$ to $a_1$ , and fixing all other elements. Its support is one orbit of length $r$ . Every finite permutation is a product of cycles with disjoint supports, uniquely up to the order of the cycles and cyclic rotation of each notation.

Single-cycle permutations

An $n$ -cycle on an $n$ -point set is transitive: every point lies in the same orbit. There are $(n-1)!$ such permutations, obtained by fixing one starting element and ordering the remaining elements around the cycle.

These are precisely the ergodic permutations for uniform probability. For $n=1$ , the identity is the one-cycle and the system is ergodic.