Lifting of generating subgroups

Abstract:

Let $\varphi \colon G\to H$ be an epimorphism of finite groups. Suppose that $G$ is generated by its subgroups $G_{1} ,\ldots ,G_{n}$ and that $H$ is generated by its subgroups $H_{1},\ldots ,H_{n}$. Furthermore, suppose that $\varphi (G_{i})$ and $H_{i}$ are conjugate, $i=1,\ldots ,n$. We prove that there exist $g_{1},\ldots ,g_{n}\in G$ such that $G_{1}^{g_{1}} ,\ldots ,G_{n}^{g_{n}}$ generate $G$ and $\varphi (G_{i}^{g_{i}})=H_{i}$, $i=1 ,\ldots ,n$.