Abelian quotients and orbit sizes of finite groups

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. It has been known for a long time that if $G$ is abelian, then $G$ has a regular orbit on $V$. In this paper we show that $G$ has an orbit of size at least $\mathrm{\mid }G\mathrm{/}{G}^{\mathrm{\prime }}\mathrm{\mid }$ on $V$. This generalizes earlier work of the authors, where the same bound was proved under the additional hypothesis that $G$ is solvable. For completely reducible modules it also strengthens the 1989 result $\mathrm{\mid }G\mathrm{/}{G}^{\mathrm{\prime }}\mathrm{\mid }<\mathrm{\mid }V\mathrm{\mid }$ by Aschbacher and Guralnick.