The following theorem is known:

Let $Y$ be a compact Riemann surface, $B \subseteq Y$ finite and $q \in Y \setminus B$. Then there exists a $1-1$ correspondence between classes of homomorphism of holomorphic maps of a given degree $d$ and branch locus $B$ and group homomorphisms $\rho:\pi_1(Y \setminus B,q) \to S_d$ (where $S_d$ is the group of permutations on $d$ elements) with transitive image (up to conjugation).

Now, assume that $\pi_1(Y \setminus B,q)$ is a free group on $n$ generators. How can I prove that there are only a finite number of group homomorphisms $\rho:\pi_1(Y \setminus B,q) \to S_d$ with transitive image (up to conjugation)? Does it suffice to say that any possible group homomorphism is determined by the images of the generators and since these images are finite ($S_d$ is finite), then they are finite? Is there any formula asserting the precise number of such group homomorphisms?


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