More formally, for the enlarged definition, a permutation of a set X, viewed as a bijective function , is called a cycle if the action on X of the subgroup generated by has at most one orbit with more than a single element. This notion is most commonly used when X is a finite set; then the largest orbit, S, is also finite. Let be any element of S, and put for any . If S is finite, there is a minimal number for which . Then , and is the permutation defined by

: for 0 <= i < k

and

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