A Solvability Criterion for Finite Groups Related to Character Degrees


Let m > 1 be a fixed positive integer. In this paper, we consider finite groups each of whose nonlinear character degrees has exactly m prime divisors. We show that such groups are solvable whenever m > 2. Moreover, we prove that if G is a non-solvable group with this property, then m = 2 and G is an extension of A7 or S7 by a solvable group.

This is a preview of subscription content, access via your institution.


  1. [1]

    D. Benjamin: Coprimeness among irreducible character degrees of finite solvable groups. Proc. Am. Math. Soc. 125 (1997), 2831–2837.

    MathSciNet  Article  Google Scholar 

  2. [2]

    M. Bianchi, D. Chillag, M. L. Lewis, E. Pacifiai: Character degree graphs that are complete graphs. Proc. Am. Math. Soc. 135 (2007), 671–676.

    MathSciNet  Article  Google Scholar 

  3. [3]

    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford, 1985.

    Google Scholar 

  4. [4]

    GAP Group: GAP — Groups, Algorithms, and Programming. A System for Computational Discrete Algebra. Version 4.7.4. Available at http://www.gap-system.org (2014).

  5. [5]

    I. M. Isaacs: Character Theory of Finite Groups. Pure and Applied Mathematics 69. Academic Press, New York, 1976.

    Google Scholar 

  6. [6]

    I. M. Isaacs, D. S. Passman: A characterization of groups in terms of the degrees of their characters II. Pac. J. Math. 24 (1968), 467–510.

    MathSciNet  Article  Google Scholar 

  7. [7]

    G. James, A. Kerber: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and Its Applications 16. Addison-Wesley, Reading, 1981.

    Google Scholar 

  8. [8]

    O. Manz: Endliche auflösbare Gruppen, deren sämtliche Charaktergrade Primzahlpotenzen sind. J. Algebra 94 (1985), 211–255. (In German.)

    MathSciNet  Article  Google Scholar 

  9. [9]

    O. Manz: Endliche nicht-auflösbare Gruppen, deren sämtliche Charaktergrade Primzahlpotenzen sind. J. Algebra 96 (1985), 114–119. (In German.)

    MathSciNet  Article  Google Scholar 

  10. [10]

    P. Schmid: Extending the Steinberg representation. J. Algebra 150 (1992), 254–256.

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Sajjad Mahmood Robati.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Miraali, B., Robati, S.M. A Solvability Criterion for Finite Groups Related to Character Degrees. Czech Math J 70, 1205–1209 (2020). https://doi.org/10.21136/CMJ.2020.0440-19

Download citation


  • non-solvable group
  • solvable group
  • character degree

MSC 2020

  • 20C15
  • 20D10