Abelian quotients and orbit sizes of linear groups

  • Thomas Michael Keller
  • Yong YangEmail author


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 


20D99 20E45 


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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.


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTexas State UniversitySan MarcosUSA
  2. 2.Key Laboratory of Group and Graph Theories and ApplicationsChongqing University of Arts and SciencesChongqingChina

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