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Relatively Maximal Subgroups of Odd Index in Symmetric Groups

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Algebra and Logic Aims and scope

Let 𝖃 be a class of finite groups which contains a group of order 2 and is closed under subgroups, homomorphic images, and extensions. We define the concept of an 𝖃-admissible diagram representing a natural number n. Associated with each n are finitely many such diagrams, and they all can be found easily. Admissible diagrams representing a number n are used to uniquely parametrize conjugacy classes of maximal 𝖃-subgroups of odd index in the symmetric group Symn, and we define the structure of such groups. As a consequence, we obtain a complete classification of submaximal 𝖃-subgroups of odd index in alternating groups.

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Notes

  1. The two statements can be viewed uniformly if we assume that painting over the upper part of any column with the number 1 above it is done conditionally. Therefore, the column can also be conceived of as being fully unpainted, cut into strips of heights λ1, . . . , λk, Ό = λk+1, with the height of the painted part equal to zero.

  2. Possibly, the only not quite trivial fact is that the same group cannot correspond to two different admissible diagrams. This follows easily from the observation that lengths of orbits of such a group S𝒟 uniquely specify the template of a diagram 𝒟, while the impossibility of cutting columns of that template in two different ways for obtaining an admissible diagram corresponding to S𝒟 is easily derived by induction from Lemma 2.9.

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Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, Russia

    A. S. Vasil’ev & D. O. Revin

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. S. Vasil’ev & D. O. Revin

  3. Novosibirsk State University, Novosibirsk, Russia

    A. S. Vasil’ev & D. O. Revin

Authors
  1. A. S. Vasil’ev
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  2. D. O. Revin
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Correspondence to A. S. Vasil’ev or D. O. Revin.

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Translated from Algebra i Logika, Vol. 61, No. 2, pp. 150-179, March-April, 2022. Russian DOI: https://doi.org/10.33048/alglog.2022.61.202.

Supported by Russian Science Foundation, project No. 19-71-10067.

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Vasil’ev, A.S., Revin, D.O. Relatively Maximal Subgroups of Odd Index in Symmetric Groups. Algebra Logic 61, 104–124 (2022). https://doi.org/10.1007/s10469-022-09680-0

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