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The Wielandt–Hartley theorem for submaximal \(\mathfrak {X}\)-subgroups

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Abstract

A nonempty class \(\mathfrak {X}\) of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We consider two definitions of submaximal \(\mathfrak {X}\)-subgroups suggested by H. Wielandt and discuss which one better suits the task of determining maximal \(\mathfrak {X}\)-subgroups. We prove that these definitions are not equivalent yet the Wielandt–Hartley theorem holds true for either definition of \(\mathfrak {X}\)-submaximality. We also give some applications of the strong version of the Wielandt–Hartley theorem.

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Notes

  1. We consider finite groups only, and from now on the term “group” will mean a “finite group”.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090

    Danila Revin, Saveliy Skresanov & Andrey Vasil’ev

  2. Novosibirsk State University, 1 Pirogova street, Novosibirsk, Russia, 630090

    Danila Revin, Saveliy Skresanov & Andrey Vasil’ev

Authors
  1. Danila Revin
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  2. Saveliy Skresanov
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  3. Andrey Vasil’ev
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Correspondence to Andrey Vasil’ev.

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Communicated by Adrian Constantin.

To the 110th anniversary of the birth of Helmut Wielandt.

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Danila Revin and Andrey Vasil’ev were supported by the program of fundamental scientific researches of the Russian Federation, Project No. 0314-2019-0001. Saveliy Skresanov was supported by Russian Foundation for Basic Research, Project No. 18-31-20011.

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Revin, D., Skresanov, S. & Vasil’ev, A. The Wielandt–Hartley theorem for submaximal \(\mathfrak {X}\)-subgroups. Monatsh Math 193, 143–155 (2020). https://doi.org/10.1007/s00605-020-01425-4

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  • DOI: https://doi.org/10.1007/s00605-020-01425-4

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