Consider a rational function $g=\frac{P(x)}{Q(x)}:S^2\to S^2$ and let $f$ be any Mobius function on $S^2$ where $S^2$ denotes Riemann sphere.
Let $p$ be a prime$\geq 2$. If $g^p=f$, then $g$ has order $1,p,\infty$ by considering the group formed by $(g)$ where multiplication is composition obviously.
Q1. Is this statement true in general?
I could show for concrete case like 5 being the case by explicit computation.
Q2. How do I show the general statement? I would love to have hints rather than solutions.