Monatshefte für Mathematik

, Volume 185, Issue 1, pp 159–162 | Cite as

A variation on a theorem of Gluck

Article
  • 76 Downloads

Abstract

Gluck proved that any finite group G has an abelian subgroup A such that |G : A| is bounded by a polynomial function of the largest degree of the complex irreducible characters of G. This improved on a previous bound of Isaacs and Passman. Moretó (J. Algebra 301:274–279, 2006) presented a variation of this result that looks at the number of prime factors and obtained an almost quadratic bound. In this note, we improve the result of Moretó to almost linear.

Keywords

Character degrees Prime divisors Arithmetic properties 

Mathematics Subject Classification

20C15 

Notes

Acknowledgements

The project is supported by a Texas State University Research Enhancement Program. I am grateful to the referee for his/her valuable suggestions which improved the manuscript.

References

  1. 1.
    Gluck, D.: The largest irreducible character degree of a finite group. Can. J. Math. 37(3), 442–451 (1985)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Isaacs, I.M.: Character Theory of Finite Groups. Dover, New York (1994)MATHGoogle Scholar
  3. 3.
    Isaacs, I.M., Passman, D.S.: Groups with representations of bounded degree. Can. J. Math 16, 299–309 (1964)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Moretó, A.: A variation on theorems of Jordan and Gluck. J. Algebra 301, 274–279 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Yang, Y.: Orbits of the actions of finite solvable groups. J. Algebra 321, 2012–2021 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2017

Authors and Affiliations

  1. 1.Department of MathematicsTexas State UniversitySan MarcosUSA

Personalised recommendations