Let G be a finite group, and let V be a completely reducible faithful finite G-module (i.e., G ⩽ GL(V), where V is a finite vector space which is a direct sum of irreducible G-submodules). 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.
abelian quotients orbits of group actions linear groups
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This work was supported by National Natural Science Foundation of China (Grant No. 11671063), a grant from the Simons Foundation (Grant No. 280770 to Thomas M. Keller), and a grant from the Simons Foundation (Grant No. 499532 to Yong Yang). The authors are also grateful to the referees for their helpful suggestions and comments.