Siberian Mathematical Journal

, Volume 53, Issue 3, pp 419–430 | Cite as

Pronormality of Hall subgroups in finite simple groups

  • E. P. Vdovin
  • D. O. Revin


We prove that the Hall subgroups of finite simple groups are pronormal. Thus we obtain an affirmative answer to Problem 17.45(a) of the Kourovka Notebook.


Hall subgroup pronormal subgroup simple group 


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  1. 1.
    Vdovin E. P., “Carter subgroups of finite almost simple groups,” Algebra and Logic, 46, No. 2, 90–119 (2007).MathSciNetCrossRefGoogle Scholar
  2. 2.
    The Kourovka Notebook, Unsolved Problems in Group Theory. Edited by V. D. Mazurov and E. I. Khukhro. 17th. ed., Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk (2010).MATHGoogle Scholar
  3. 3.
    Vdovin E. P. and Revin D. O., “Theorems of Sylow type,” Russian Math. Surveys, 66, No. 5, 829–870 (2011).MATHCrossRefGoogle Scholar
  4. 4.
    Vdovin E. P. and Revin D. O., “A conjugacy criterion for Hall subgroups in finite groups,” Siberian Math. J., 51, No. 3, 402–409 (2010).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Revin D. O., “Around a conjecture of P. Hall,” Sib. Electronic Math. Reports, 6, 366–380 (2009).MathSciNetGoogle Scholar
  6. 6.
    Conway J. H., Curtis R. T., Norton S. P., Parker R. A., and Wilson R. A., Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press, Oxford (1985).MATHGoogle Scholar
  7. 7.
    Hall P., “Theorems like Sylow’s,” Proc. London Math. Soc., 6, No. 22, 286–304 (1956).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Chunikhin S. A., “On Sylow properties of finite groups,” Dokl. Akad. Nauk SSSR, 73, No. 1, 29–32 (1950).MATHGoogle Scholar
  9. 9.
    Gross F., “Conjugacy of odd order Hall subgroups,” Bull. London Math. Soc., 19, No. 4, 311–319 (1987).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Revin D. O. and Vdovin E. P., “Hall subgroups of finite groups,” in: Ischia Group Theory 2004: Proc. Conf. in Honour of Marcel Herzog (Naples (Italy), March 31–April 3, 2004), Amer. Math. Soc., Providence, 2006, pp. 229–265 (Contemp. Math.; 402).Google Scholar
  11. 11.
    Kleidman P. B. and Liebeck M., The Subgroup Structure of the Finite Classical Groups, Cambridge Univ. Press, Cambridge (1990).MATHCrossRefGoogle Scholar
  12. 12.
    Gorenstein D., Lyons R., and Solomon R., The Classification of the Finite Simple Groups, Amer. Math. Soc., Providence (1994) (Math. Surveys and Monogr.; V. 40(1)).MATHGoogle Scholar
  13. 13.
    Carter R. W., Simple Groups of Lie Type, John Wiley and Sons, London (1972).MATHGoogle Scholar
  14. 14.
    Kondrat’ev A. S., “Subgroups of finite Chevalley groups,” Russian Math. Surveys, 41, No. 1, 65–118 (1986).MATHCrossRefGoogle Scholar
  15. 15.
    Carter R. W., Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, John Wiley and Sons, New York (1985).MATHGoogle Scholar
  16. 16.
    Kondrat’ev V. A., “Normalizers of the Sylow 2-subgroups in finite simple groups,” Math. Notes, 78, No. 3, 338–346 (2005).MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Thompson J. G., “Hall subgroups of the symmetric groups,” J. Combin. Theory Ser. A, 1, No. 2, 271–279 (1966).MATHCrossRefGoogle Scholar
  18. 18.
    Revin D. O., “The D π-property in a class of finite groups,” Algebra and Logic, 41, No. 3, 187–206 (2002).MathSciNetCrossRefGoogle Scholar
  19. 19.
    Revin D. O., “Hall π-subgroups of finite Chevalley groups whose characteristic belongs to π,” Siberian Adv. in Math., 9, No. 2, 25–71 (1999).MathSciNetMATHGoogle Scholar
  20. 20.
    Revin D. O. and Vdovin E. P., “On the number of classes of conjugate Hall subgroups in finite simple groups,” J. Algebra, 324, No. 12, 3614–3652 (2010).MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Aschbacher M., “On finite groups of Lie type and odd characteristic,” J. Algebra, 66, No. 1, 400–424 (1980).MathSciNetMATHCrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics and Novosibirsk State UniversityNovosibirskRussia

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