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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 sizeat least $|G/G'|$ on $V$. This generalizes earlier work of the authors, wherethe same bound was proved under the additional hypothesis that $G$ is solvable.For completely reducible modules it also strengthens the 1989 result$|G/G'|<|V|$ by Aschbacher and Guralnick.

Abelian quotients and orbit sizes of finite groups