Inventiones mathematicae

, Volume 170, Issue 1, pp 33–101 | Cite as

A reduction theorem for the McKay conjecture

  • I.M. Isaacs
  • Gunter Malle
  • Gabriel Navarro


The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL2(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 e or q=3 e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J 1, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.


Simple Group Maximal Torus Irreducible Character Linear Character Semisimple Element 
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  1. 1.
    Alperin, J.L.: The main problem of block theory. In: Proceedings of the Conference on Finite Groups, Park City, Utah, 1975, pp. 341–356. Academic Press, New York (1976)Google Scholar
  2. 2.
    Carter, R.W.: Finite Groups of Lie Type. Wiley-Interscience, New York (1985)zbMATHGoogle Scholar
  3. 3.
    Dade, E.C.: Counting characters in blocks I. Invent. Math. 109, 187–210 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dade, E.C.: Counting characters in blocks II. J. Reine Angew. Math. 448, 97–190 (1994)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Eaton, C.: Dade’s inductive conjecture for the Ree groups of type G 2 in the defining characteristic. J. Algebra 226, 614–620 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Eaton, C.W., Robinson, G.R.: On a minimal counterexample to Dade’s projective conjecture. J. Algebra 249(2), 453–462 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Green, J.A., Lehrer, G.I., Lusztig, G.: On the degrees of certain group characters. Q. J. Math. Oxf. II Ser. 27, 1–4 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Huppert, B., Blackburn, N.: Finite Groups. III. Grundlehren Math. Wiss., vol. 243. Springer, Berlin New York (1982)Google Scholar
  9. 9.
    Isaacs, I.M.: Character Theory of Finite Groups. Dover, New York (1994)zbMATHGoogle Scholar
  10. 10.
    Isaacs, I.M.: Algebra: A Graduate Course. Brooks-Cole, Pacific Grove (1994)Google Scholar
  11. 11.
    Isaacs, I.M., Navarro, G.: New refinements of the McKay conjecture for arbitrary finite groups. Ann. Math. (2) 156, 333–344 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lusztig, G.: On the representations of reductive groups with disconnected centre. Astérisque 168, 157–166 (1988)Google Scholar
  13. 13.
    Malle, G.: The inductive McKay condition for simple groups not of Lie type. (to appear in Comm. Algebra.)Google Scholar
  14. 14.
    Malle, G.: Height 0 characters of finite groups of Lie type. (submitted 2006)Google Scholar
  15. 15.
    Murai, M.: A remark on the Alperin–McKay conjecture. J. Math. Kyoto Univ. 44, 245–254 (2004)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Navarro, G.: The McKay conjecture and Galois automorphisms. Ann. Math. 160, 1129–1140 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Okuyama, T., Wajima, M.: Character correspondence and p-blocks of p-solvable groups. Osaka J. Math. 17, 801–806 (1980)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Robinson, G.R.: Dade’s projective conjecture for p-solvable groups. J. Algebra 229, 234–248 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Späth, B.: Die McKay Vermutung für quasi-einfache Gruppen vom Lie-Typ. Doctoral thesis, Technische Universität Kaiserslautern (2007)Google Scholar
  20. 20.
    Suzuki, M.: On a class of doubly transitive groups. Ann. Math. 75, 105–145 (1962)CrossRefGoogle Scholar
  21. 21.
    Turull, A.: Strengthening the McKay conjecture to include local fields and local Schur indices. (to appear)Google Scholar
  22. 22.
    Ward, H.N.: On Ree’s series of simple groups. Trans. Am. Math. Soc. 121, 62–89 (1966)zbMATHCrossRefGoogle Scholar
  23. 23.
    Walter, J.H.: The characterization of finite groups with abelian Sylow 2-subgroups. Ann. Math. 89, 405–514 (1969)CrossRefGoogle Scholar
  24. 24.
    Wilson, R.A.: The McKay conjecture is true for the sporadic simple groups. J. Algebra 207, 294–305 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Wolf, T.R.: Variations on McKay’s character degree conjecture. J. Algebra 135, 123–138 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of WisconsinMadisonUSA
  2. 2.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  3. 3.Departament d’Àlgebra, Facultat de MatemàtiquesUniversitat de ValènciaBurjassotSpain

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