Paper 3, Section II, D

In the group of Möbius maps, what is the order of the Möbius map $z \mapsto \frac{1}{z}$ ? What is the order of the Möbius map $z \mapsto \frac{1}{1-z}$ ?

Prove that every Möbius map is conjugate either to a map of the form $z \mapsto \mu z$ (some $\mu \in \mathbb{C}$ ) or to the $\operatorname{map} z \mapsto z+1$. Is $z \mapsto z+1$ conjugate to a map of the form $z \mapsto \mu z ?$

Let $f$ be a Möbius map of order $n$, for some positive integer $n$. Under the action on $\mathbb{C} \cup\{\infty\}$ of the group generated by $f$, what are the various sizes of the orbits? Justify your answer.

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