Finite-order Möbius orbit sizes ($|\mathcal{O}(z)|\in \{1,n\}$)

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.