On Galois group of factorized covers of curves
Summary: Let \(\mathcal{Y} \mathop{\rightarrow}\limits^{\psi}\mathcal{X} \mathop{\rightarrow}\limits^{\varphi} \mathbb{P}^{1}\) 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 \( \mathfrak{G}(\phi\circ\psi) \cong ({\mathbb Z}_{n_1} \times {\mathbb Z}_{n_2} \times \cdots \times {\mathbb Z}_{ns})^{m-1} \rtimes {\mathbf S}_m\). 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 \(\phi\circ\psi.\)
scientific article; zbMATH DE number 7023581
scientific article