On Galois group of factorized covers of curves

Abstract

Let $\mathcal{Y}\stackrel{\psi }{\to }\mathcal{X}\stackrel{\phi }{\to }{\mathbb{P}}^1$ be a sequence of covers of compact Riemann surfaces. In this work we study and completely characterize the Galois group $\mathfrak{G}(\phi \circ \psi )$ of $\phi \circ \psi$ under the following properties: $\phi$ is a simple cover of degree $m$ and $\psi$ is a Galois unramified cover of degree $n$ with abelian Galois group of type $(n_1,n_2,\dots ,n_s)$.

We prove that $\mathfrak{G}(\phi \circ \psi )\cong ({\mathbb{Z}}_{n_1}\times {\mathbb{Z}}_{n_2}\times \cdots \times {\mathbb{Z}}_{n_s}{)}^{m-1}⋊{\mathbf{S}}_m$. Furthermore, we give a natural geometric generator system of $\mathfrak{G}(\phi \circ \psi )$ obtained by studying the action on the compact Riemann surface $\mathcal{Z}$ associated to the Galois closure of $\phi \circ \psi .$