Let $f: R \rightarrow S$ be a non-constant holomorphic map between compact connected Riemann surfaces and let $B \subset S$ denote the set of branch points. Show that the map $f: R \backslash f^{-1}(B) \rightarrow S \backslash B$ is a regular covering map.

Given $w \in S \backslash B$ and a closed curve $\gamma$ in $S \backslash B$ with initial and final point $w$, explain how this defines a permutation of the (finite) set $f^{-1}(w)$. Show that the group $H$ obtained from all such closed curves is a transitive subgroup of the full symmetric group of the fibre $f^{-1}(w)$.

Find the group $H$ for $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ where $f(z)=z^{3} /\left(1-z^{2}\right)$.