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arXiv · 1407.6436

Abelian quotients and orbit sizes of finite groups

Thomas Michael Keller·Yong Yang
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Abstract

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 $|G/G'|$ 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 $|G/G'|<|V|$ by Aschbacher and Guralnick.

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Thomas Michael Keller, Yong Yang. 2014-07-24. Abelian quotients and orbit sizes of finite groups. https://arxiv.org/abs/1407.6436

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