Disjoint cycle decomposition ($\sigma ={\gamma }_1\cdots {\gamma }_r$)

A disjoint cycle decomposition writes a permutation as a product of disjoint permutation cycles. Repeatedly apply the permutation to a point until it returns to that point; these successive points form one cycle. Repeat on points not yet used. Finiteness and injectivity ensure that each process returns to its starting point. This partitions the underlying finite set into cycles. The decomposition is unique up to reordering the cycles, cyclically rotating the entries within each cycle, and optionally omitting fixed-point one-cycles. Include the one-cycles when specifying cycle type.