= Cycle of a permutation
{title2=$x,\sigma x,\ldots,\sigma^{k-1}x$}
= Cycles of a permutation
{synonym}
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 $x$.
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