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Abelian quotients and orbit sizes of linear groups

  • Thomas Michael Keller
  • Yong YangEmail author
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Abstract

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.

Keywords

abelian quotients orbits of group actions linear groups 

MSC(2010)

20D99 20E45 

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Notes

Acknowledgements

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.

References

  1. 1.
    Aschbacher M, Guralnick R M. On abelian quotients of primitive groups. Proc Amer Math Soc, 1989, 107: 89–95MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Conway J H, Curtis R T, Norton S P, et al. Atlas of Finite Groups. New York: Oxford University Press, 1985zbMATHGoogle Scholar
  3. 3.
    Feit W. On large Zsigmondy primes. Proc Amer Math Soc, 1988, 102: 29–36MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    GAP. The GAP Group, GAP-Groups, Algorithms, and Programming. Version 4.3, 2002Google Scholar
  5. 5.
    Gorenstein D. Finite Groups. New York: Harper and Row, 1968Google Scholar
  6. 6.
    Glasby S P, Lübeck F, Niemeyer A C, et al. Primitive prime divisors and the nth cyclotomic polynomial. J Aust Math Soc, 2017, 102: 122–135MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gorenstein D, Lyons R, Solomon R. The Classification of the Finite Simple Groups, No. 2. Providence: Amer Math Soc, 1996zbMATHGoogle Scholar
  8. 8.
    Keller T M, Yang Y. Abelian quotients and orbit sizes of solvable linear groups. Israel J Math, 2016, 211: 23–44MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Manz O, Wolf T R. Representations of Solvable Groups. Cambridge: Cambridge University Press, 1993CrossRefzbMATHGoogle Scholar
  10. 10.
    Niemeyer A C, Praeger C E. A recognition algorithm for classical groups over finite fields. Proc London Math Soc (3), 1998,, 77: 117–169MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Qian G, Yang Y. Permutation characters in finite solvable groups. Comm Algebra, 2018, 46: 167–175MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sambale B. Cartan matrices and Brauer's k(B)-conjecture III. Manuscripta Math, 2015, 146: 505–518MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wilson R A. The Finite Simple Groups. Graduate Texts in Mathematics, 251. London: Springer-Verlag London, 2009CrossRefzbMATHGoogle Scholar

Copyright information

© 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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