= Finite-order Möbius orbit sizes
{title2=$|\mathcal O(z)|\in\{1,n\}$}
A nonidentity <Möbius transformation> of finite order $n$ is conjugate to multiplication by a primitive $n$th <root of unity>. Zero and infinity are fixed; for any other point, a return after $k$ iterates requires $\mu^k=1$, whose least positive solution is $n$. Thus its only orbit sizes are one and $n$. For the identity, every orbit has size one. This is a direct orbit computation, independent of an appeal to the orbit-stabilizer theorem.
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