Advertisement

Siberian Mathematical Journal

, Volume 58, Issue 3, pp 467–475 | Cite as

On the pronormality of subgroups of odd index in finite simple symplectic groups

  • A. S. Kondrat’evEmail author
  • N. V. Maslova
  • D. O. Revin
Article

Abstract

A subgroup H of a group G is pronormal if the subgroups H and H g are conjugate in 〈H,H g 〉 for every gG. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSL n (q), PSU n (q), E 6(q), 2 E 6(q), where in all cases q is odd and n is not a power of 2, and P Sp2n (q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp2n (q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.

The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp2n (q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2 m or 2 m (22k +1), this group has a nonpronormal subgroup of odd index. If n = 2 m , then we show that all subgroups of P Sp2n (q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp2n (q) is still open when n = 2 m (22k + 1) and q ≡ ±3 (mod 8).

Keywords

finite group simple group symplectic group pronormal subgroup odd index 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Vdovin E. P. and Revin D. O., “Pronormality of Hall subgroups in finite simple groups,” Sib. Math. J., vol. 53, no. 3, 419–430 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kondrat’ev A. S., Maslova N. V., and Revin D. O., “On the pronormality of subgroups of odd index in finite simple groups,” Sib. Math. J., vol. 56, no. 6, 1101–1107 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kondrat’ev A. S., Maslova N. V., and Revin D. O., “A pronormality criterion for supplements to abelian normal subgroups,” Proc. Steklov Inst. Math., vol. 296, suppl. 1, S145–S150 (2017).CrossRefzbMATHGoogle Scholar
  4. 4.
    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).zbMATHGoogle Scholar
  5. 5.
    Kleidman P. B. and Liebeck M., The Subgroup Structure of the Finite Classical Groups, Cambridge Univ. Press, Cambridge (1990).CrossRefzbMATHGoogle Scholar
  6. 6.
    Kondrat’ev A. S., “Normalizers of the Sylow 2-subgroups in finite simple groups,” Math. Notes, vol. 78, no. 3, 338–346 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mazurov V. D., “On the set of orders of elements of a finite group,” Algebra and Logic, vol. 33, no. 1, 49–55 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huppert B., Endliche Gruppen. I, Springer-Verlag, Berlin (1967).CrossRefzbMATHGoogle Scholar
  9. 9.
    Isaacs I. M., Character Theory of Finite Groups, Academic Press, New York (1976).zbMATHGoogle Scholar
  10. 10.
    Isaacs I. M., Finite Group Theory, Amer. Math. Soc., Providence (2008).CrossRefzbMATHGoogle Scholar
  11. 11.
    Carter R. and Fong P., “The Sylow 2-subgroups of the finite classical groups,” J. Algebra, vol. 1, no. 1, 139–151 (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Maslova N. V., “Classification of maximal subgroups of odd index in finite simple classical groups,” Proc. Steklov Inst. Math., vol. 267, suppl. 1, S164–S183 (2009).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • A. S. Kondrat’ev
    • 1
    Email author
  • N. V. Maslova
    • 1
  • D. O. Revin
    • 2
    • 3
  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Federal UniversityEkaterinburgRussia
  2. 2.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  3. 3.Department of MathematicsUniversity of Science and Technology of ChinaHefeiP. R. China

Personalised recommendations