= Order of a permutation
{title2=$\operatorname{ord}(\pi)$}
= permutation order
{synonym}
The <order of a permutation> $\pi$ is the least positive <integer> $m$ for which $\pi^m$ is the <identity permutation>. If its <disjoint permutation cycles> have lengths $\ell_1,\ldots,\ell_r$, then
$$
\operatorname{ord}(\pi)=\operatorname{lcm}(\ell_1,\ldots,\ell_r).
$$
Indeed a point in an $\ell$-cycle returns exactly after a multiple of $\ell$ steps, so all points return exactly at common multiples. The <permutation> $(1\,2\,3)(4\,5)$ has order six, while its five moved labels do not by themselves determine its order. The <identity permutation> has order one.
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