Monatshefte für Mathematik

, Volume 166, Issue 3–4, pp 559–577 | Cite as

Simple exceptional groups of Lie type are determined by their character degrees

Article

Abstract

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let \({{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}}\) be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and \({{\rm cd}(S)\subseteq {\rm cd}(H)}\) then S must be isomorphic to H. As a consequence, we show that if G is a finite group with \({{\rm X}_1(G)\subseteq {\rm X}_1(H)}\) then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.

Keywords

Character degrees Simple exceptional group 

Mathematics Subject Classification (2000)

Primary 20C15 20D05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berkovich, Y., Zhmud́, E.: Characters of finite groups. Part 1. In: Translations of Mathematical Monographs, vols. 172, 181. AMS, Providence (1997)Google Scholar
  2. 2.
    Carter R.: Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. Wiley, New York (1985)MATHGoogle Scholar
  3. 3.
    Chang, B., Ree, R.: The characters of G 2(q). In: Symposia Mathematica, vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome), pp. 395–413 (1972)Google Scholar
  4. 4.
    Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of Finite Groups. Oxford University Press, Eynsham (1985)MATHGoogle Scholar
  5. 5.
    Deriziotis D.I., Michler G.O.: Character table and blocks of finite simple triality groups 3 D 4(q). Trans. Am. Math. Soc. 303(1), 39–70 (1987)MathSciNetMATHGoogle Scholar
  6. 6.
    Enomoto H.: The characters of the finite symplectic group Sp(4, q), q = 2f. Osaka J. Math. 9, 75–94 (1972)MathSciNetMATHGoogle Scholar
  7. 7.
    Enomoto H.: The characters of the finite Chevalley group G 2(q), q = 3f. Jpn. J. Math. (N.S.) 2(2), 191–248 (1976)MathSciNetGoogle Scholar
  8. 8.
    Enomoto H., Yamada H.: The characters of G 2(2n). Jpn. J. Math. (N.S.) 12(2), 325–377 (1986)MathSciNetMATHGoogle Scholar
  9. 9.
    The GAP Group, GAP-Groups: Algorithms, and Programming, Version 4.4.10. http://www.gap-system.org (2007)
  10. 10.
    Hagie M.: The prime graph of a sporadic simple group. Commun. Algebra 31(9), 4405–4424 (2003)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Huppert B.: Some simple groups which are determined by the set of their character degrees. I. Illinois J. Math. 44(4), 828–842 (2000)MathSciNetMATHGoogle Scholar
  12. 12.
    Isaacs, M.: Character Theory of Finite Groups. Corrected reprint of the 1976 original [Academic Press, New York]. AMS Chelsea Publishing, Providence (2006)Google Scholar
  13. 13.
    Kleidman, P., Liebeck, M.W.: The subgroup structure of the finite classical groups. In: LMS Lecture Note Series, vol. 129. Cambridge University Press (1990)Google Scholar
  14. 14.
    Lübeck F.: Smallest degrees of representations of exceptional groups of Lie type. Commun. Algebra 29(5), 2147–2169 (2001)MATHCrossRefGoogle Scholar
  15. 15.
  16. 16.
    Malle G., Moretó A.: Nonsolvable groups with few character degrees. J. Algebra 294(1), 117–126 (2005)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Malle G., Zalesskii A.: Prime power degree representations of quasi-simple groups. Arch. Math. (Basel) 77(6), 461–468 (2001)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Maróti A.: Bounding the number of conjugacy classes of a permutation group. J. Group Theory 8(3), 273–289 (2005)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Mazurov, V.D., Khukhro, E.I. (eds.): Unsolved Problems in Group Theory. The Kourovka Notebook, No. 16. Inst. Mat. Sibirsk. Otdel. Akad. Novosibirsk (2006)Google Scholar
  20. 20.
    Seitz G.: Cross-characteristic embeddings of finite groups of Lie type, finite and algebraic. Proc. LMS 60, 166–200 (1990)MathSciNetMATHGoogle Scholar
  21. 21.
    Shahabi, M.A., Mohtadifar, H.: The characters of finite projective symplectic group PSp(4, q). In: Groups St. Andrews 2001 in Oxford, vol. II, pp. 496–527. London Math. Soc. Lecture Note Ser., vol. 305. Cambridge University Press, Cambridge (2003)Google Scholar
  22. 22.
    Simpson W.A., Frame J.S.: The character tables for SL(3, q), SU(3, q 2), PSL(3, q), PSU(3, q 2) . Can. J. Math. 25, 486–494 (1973)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Suzuki M.: On a class of doubly transitive groups. Ann. Math. (2) 75, 105–145 (1962)MATHCrossRefGoogle Scholar
  24. 24.
    Tong-Viet, H.P.: Alternating and sporadic simple groups are determined by their character degrees. Algebr. Represent. Theory. doi:10.1007/s10468-010-9247-1 (2010)
  25. 25.
    Tong-Viet, H.P.: Simple classical groups of Lie type are determined by their character degrees. PreprintGoogle Scholar
  26. 26.
    Tong-Viet, H.P.: Symmetric groups are determined by their character degrees. J. Algebra. doi:10.1016/j.jalgebra.2010.11.018 (2011)
  27. 27.
    Ward H.N.: On Ree’s series of simple groups. Trans. Am. Math. Soc. 121, 62–89 (1966)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of KwaZulu-NatalPietermaritzburgSouth Africa

Personalised recommendations