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A reduction theorem for the McKay conjecture

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Abstract

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=2e or q=3e 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.

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References

  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. Carter, R.W.: Finite Groups of Lie Type. Wiley-Interscience, New York (1985)

    MATH  Google Scholar 

  3. Dade, E.C.: Counting characters in blocks I. Invent. Math. 109, 187–210 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dade, E.C.: Counting characters in blocks II. J. Reine Angew. Math. 448, 97–190 (1994)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Eaton, C.W., Robinson, G.R.: On a minimal counterexample to Dade’s projective conjecture. J. Algebra 249(2), 453–462 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  8. Huppert, B., Blackburn, N.: Finite Groups. III. Grundlehren Math. Wiss., vol. 243. Springer, Berlin New York (1982)

    Google Scholar 

  9. Isaacs, I.M.: Character Theory of Finite Groups. Dover, New York (1994)

    MATH  Google Scholar 

  10. Isaacs, I.M.: Algebra: A Graduate Course. Brooks-Cole, Pacific Grove (1994)

    Google Scholar 

  11. Isaacs, I.M., Navarro, G.: New refinements of the McKay conjecture for arbitrary finite groups. Ann. Math. (2) 156, 333–344 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lusztig, G.: On the representations of reductive groups with disconnected centre. Astérisque 168, 157–166 (1988)

    Google Scholar 

  13. Malle, G.: The inductive McKay condition for simple groups not of Lie type. (to appear in Comm. Algebra.)

  14. Malle, G.: Height 0 characters of finite groups of Lie type. (submitted 2006)

  15. Murai, M.: A remark on the Alperin–McKay conjecture. J. Math. Kyoto Univ. 44, 245–254 (2004)

    MATH  MathSciNet  Google Scholar 

  16. Navarro, G.: The McKay conjecture and Galois automorphisms. Ann. Math. 160, 1129–1140 (2004)

    Article  MathSciNet  Google Scholar 

  17. Okuyama, T., Wajima, M.: Character correspondence and p-blocks of p-solvable groups. Osaka J. Math. 17, 801–806 (1980)

    MATH  MathSciNet  Google Scholar 

  18. Robinson, G.R.: Dade’s projective conjecture for p-solvable groups. J. Algebra 229, 234–248 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Späth, B.: Die McKay Vermutung für quasi-einfache Gruppen vom Lie-Typ. Doctoral thesis, Technische Universität Kaiserslautern (2007)

  20. Suzuki, M.: On a class of doubly transitive groups. Ann. Math. 75, 105–145 (1962)

    Article  Google Scholar 

  21. Turull, A.: Strengthening the McKay conjecture to include local fields and local Schur indices. (to appear)

  22. Ward, H.N.: On Ree’s series of simple groups. Trans. Am. Math. Soc. 121, 62–89 (1966)

    Article  MATH  Google Scholar 

  23. Walter, J.H.: The characterization of finite groups with abelian Sylow 2-subgroups. Ann. Math. 89, 405–514 (1969)

    Article  Google Scholar 

  24. Wilson, R.A.: The McKay conjecture is true for the sporadic simple groups. J. Algebra 207, 294–305 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wolf, T.R.: Variations on McKay’s character degree conjecture. J. Algebra 135, 123–138 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Mathematics Department, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706, USA

    I.M. Isaacs

  2. Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, Kaiserslautern, 67653, Germany

    Gunter Malle

  3. Departament d’Àlgebra, Facultat de Matemàtiques, Universitat de València, Burjassot, València, 46100, Spain

    Gabriel Navarro

Authors
  1. I.M. Isaacs
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  2. Gunter Malle
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  3. Gabriel Navarro
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Correspondence to I.M. Isaacs.

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Isaacs, I., Malle, G. & Navarro, G. A reduction theorem for the McKay conjecture. Invent. math. 170, 33–101 (2007). https://doi.org/10.1007/s00222-007-0057-y

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  • DOI: https://doi.org/10.1007/s00222-007-0057-y

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