Past exam of the geometry of surfaces course of the University of Oxford / 2023 / 2 / c / Solution

In a cyclic group of prime order $p$, a stabilizer subgroup has order either one or $p$. The latter case consists exactly of points fixed by all of $G$. Thus the $n$ fixed points each have ramification index of a holomorphic map $p$, and every other point has index one. Applying the Riemann-Hurwitz formula with degree of a holomorphic map $p$ givesEquivalently,The count $n$ concerns fixed points upstairs. Each is its own group orbit, so their images are also $n$ distinct branch values of a holomorphic map.