Skip to main content
SpringerLink
Log in

Finite groups with an irreducible character of large degree

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

Abstract

Let G be a finite group and d the degree of a complex irreducible character of G, then write |G| = d(d + e) where e is a nonnegative integer. We prove that |G| ≤ e 4−e 3 whenever e > 1. This bound is best possible and improves on several earlier related results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berkovich Y.: Groups with few characters of small degrees. Isr. J. Math. 110, 325–332 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Burkett S.T., Nguyen H.N.: Conjugacy classes of small sizes in the linear and unitary groups. J. Group Theory 16, 851–874 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carter, R.W.: Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Wiley, New York (1985)

  5. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)

    MATH  Google Scholar 

  6. Cossey J.P., Halasi Z., Maróti A., Nguyen H.N.: On a conjecture of Gluck. Math. Z. 279, 1067–1080 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Digne, F., Michel, J.: Representations of finite groups of Lie type. Lond. Math. Soc. Stud. Texts 21 (1991)

  8. Durfee C., Jensen S.: A bound on the order of a group having a large character degree. J. Algebra 338, 197–206 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feit W.: Extending Steinberg characters, Linear algebraic groups and their representations. Contemp. Math. 153, 1–9 (1993)

    Article  MathSciNet  Google Scholar 

  10. Frame J.S., Robinson G.B., Thrall R.M.: The hook graphs of the symmetric group. Can. J. Math. 6, 316–324 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gagola S.M. Jr.: Characters vanishing on all but two conjugacy classes. Pac. J. Math. 109, 363–385 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gluck D.: The largest irreducible character degree of a finite group. Can. J. Math. 37, 442–451 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Goldstein D., Guralnick R.M., Lewis M.L., Moretó A., Navarro G., Tiep P.H.: Groups with exactly one irreducible character of degree divisible by p. Algebra Number Theory 8, 397–428 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Halasi, Z., Hannusch, C., Nguyen, H.N.: The largest character degrees of the symmetric and alternating groups. Proc. Am. Math. Soc. arXiv:1410.3055 (to appear)

  15. Isaacs I.M.: Character Theory of Finite Groups. AMS Chelsea, Providence (2006)

    MATH  Google Scholar 

  16. Isaacs I.M.: Bounding the order of a group with a large character degree. J. Algebra 348, 264–275 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. James, G.D.: The Representation Theory of the Symmetric Groups. Lecture Notes in Mathematics. Springer, Berlin (1978)

  18. Kuisch E., Waall R.W.: Homogeneous character induction. J. Algebra 149, 454–471 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Larsen M., Malle G., Tiep P.H.: The largest irreducible representations of simple groups. Proc. Lond. Math. Soc. 106, 65–96 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lewis M.L.: Bounding group orders by large character degrees: a question of Snyder. J. Group Theory 17, 1081–1116 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lewis M.L.: Camina pairs that are not p-closed. Isr. J. Math. doi:10.1007/s11856-014-1126-8

  22. Lübeck F.: Data for Finite Groups of Lie Type and Related Algebraic Groups. http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/index.html

  23. Manz, O., Wolf, T.R.: Representations of Solvable Groups. London Mathematical Society Lecture Note Series, vol. 185. Cambridge University Press, Cambridge (1993)

  24. Mattarei S.: On character tables of wreath products. J. Algebra 175, 157–178 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. McKay J.: The largest degrees of irreducible characters of the symmetric group. Math. Comput. 30, 624–631 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. Meyn H., Götz W.: Self-reciprocal polynomials over finite fields, Séminaire Lotharingien de Combinatoire. Publ. Inst. Rech. Math. Av. 413, 82–90 (1990)

    Google Scholar 

  27. Moretó A., Nguyen H.N.: On the average character degree of finite groups. Bull. Lond. Math. Soc. 46, 454–462 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nguyen H.N.: Low-dimensional complex characters of the symplectic and orthogonal groups. Commun. Algebra 38, 1157–1197 (2010)

    Article  MATH  Google Scholar 

  29. Passman D.S.: Permutation Groups, pp. vi+152. Dover, Mineola (2012)

    MATH  Google Scholar 

  30. Seitz G.M.: Cross-characteristic embeddings of finite groups of Lie type. Proc. Lond. Math. Soc. 60, 166–200 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Snyder N.: Groups with a character of large degree. Proc. Am. Math. Soc. 136, 1893–1903 (2008)

    Article  MATH  Google Scholar 

  32. Tiep P.H., Zalesskií A.E.: Unipotent elements of finite groups of Lie type and realization fields of their complex representations. J. Algebra 271, 327–390 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. White D.L.: Character degrees of extensions of PSL2(q) and SL2(q). J. Group Theory 16, 1–33 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Akron, Akron, OH, 44325, USA

    Nguyen Ngoc Hung

  2. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    Mark L. Lewis

  3. Department of Mathematical and Computer Sciences, Metropolitan State University of Denver, Denver, CO, 80217, USA

    Amanda A. Schaeffer Fry

Authors
  1. Nguyen Ngoc Hung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark L. Lewis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amanda A. Schaeffer Fry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nguyen Ngoc Hung.

Additional information

Nguyen N. Hung is partially supported by the NSA Young Investigator Grant #H98230-14-1-0293 and a Faculty Scholarship Award from the Buchtel College of Arts and Sciences, The University of Akron.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hung, N.N., Lewis, M.L. & Schaeffer Fry, A.A. Finite groups with an irreducible character of large degree. manuscripta math. 149, 523–546 (2016). https://doi.org/10.1007/s00229-015-0793-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-015-0793-z

Mathematics Subject Classification

Search

Navigation