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Doklady Mathematics

, Volume 98, Issue 2, pp 405–408 | Cite as

On Pronormal Subgroups in Finite Simple Groups

  • A. S. Kondrat’ev
  • N. V. Maslova
  • D. O. Revin
Mathematics
  • 3 Downloads

Abstract

A subgroup H of a group G is called pronormal if, for any element g of G, the subgroups H and Hg are conjugate in the subgroup they generate. Some problems in the theory of permutation groups and combinatorics have been solved in terms of pronormality, and the characterization of pronormal subgroups in finite groups is a problem of importance for applications of group theory. A task of special interest is the study of pronormal subgroups in finite simple groups and direct products of such groups. In 2012 E.P. Vdovin and D.O. Revin conjectured that the subgroups of odd index in all finite simple groups are pronormal. We disproved this conjecture in 2016. Accordingly, a natural task is to classify finite simple groups in which the subgroups of odd index are pronormal. This paper completes the description of finite simple groups whose Sylow 2-subgroups contain their centralizers in the group and the subgroups of odd index in which are pronormal.

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • A. S. Kondrat’ev
    • 1
  • N. V. Maslova
    • 1
  • D. O. Revin
    • 2
  1. 1.Krasovskii Institute of Mathematics and Mechanics, Ural BranchRussian Academy of SciencesYekaterinburgRussia
  2. 2.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia

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