Möbius transformations of finite order I saw this question, which brought up the question: can we classify all the Möbius transformations (with complex coefficients) of finite order? In particular, do these only consist of the rotations? I sense symbolic manipulation isn't the way to go about this—and indeed didn't get anywhere with this—and would appreciate some insight.

Every Möbius transformation which is not the identity is conjugate to either a translation
$$
 z \to z + a \, \quad (a \in \Bbb C, a \ne 0)
$$
or a complex-linear map (rotation/dilation)
$$
 z \to \lambda z \, \quad (\lambda \in \Bbb C, \lambda \ne 0, 1)
$$
depending on whether it has one or two fixed points. Translations do not have finite order, and rotations if and only if $\lambda$ is some root of unity.
It follows that all Möbius transformation of finite order are of the form
$$
 T(z) = S^{-1}(\lambda S(z))
$$
with some Möbius transformation $S$ and $\lambda = e^{2 \pi i k/n} \ne 1$ for some integers $k, n$.
