 Cycle of a permutation

For a permutation $\sigma$ of a finite set, the cycle containing $x$ is its orbit under the iterates of $\sigma$. Its length $k$ is the least positive integer with ${\sigma }^{k}x=x$; finiteness and invertibility guarantee such a return. Two cycles that meet are the same, because any iterate of a common point belongs to both. Thus the cycles partition the set. In cycle notation, the displayed cycle sends each listed element to the next and the last one back to $x$.