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Maximal and Submaximal 𝔛-Subgroups

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Algebra and Logic Aims and scope

Let 𝔛 be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal 𝔛-subgroup if there exists an isomorphic embedding ϕ : G ↪ G* of G into some finite group G* under which Gϕ is subnormal in G* and Hϕ = K ∩ Gϕ for some maximal 𝔛-subgroup K of G*. In the case where 𝔛 coincides with the class of all π-groups for some set π of prime numbers, submaximal 𝔛-subgroups are called submaximal π-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal π-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal 𝔛- and π-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal 𝔛-subgroups are conjugate in a finite group G in which all maximal 𝔛-subgroups are conjugate?

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Authors and Affiliations

  1. University of Science and Technology of China, Hefei, 230026, P.R. China

    W. Guo

  2. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    D. O. Revin

  3. Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090, Russia

    D. O. Revin

  4. University of Science and Technology of China, Hefei, 230026, P.R. China

    D. O. Revin

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  1. W. Guo
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Correspondence to W. Guo.

Additional information

Translated from Algebra i Logika, Vol. 57, No. 1, pp. 14-42, January-February, 2018.

Supported by the NNSF of China, grant No. 11771409.

Supported by Chinese Academy of Sciences President’s International Fellowship Initiative (grant No. 2016-VMA078) and by SB RAS Fundamental Research Program I.1.1 (project No. 0314-2016-0001).

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Guo, W., Revin, D.O. Maximal and Submaximal 𝔛-Subgroups. Algebra Logic 57, 9–28 (2018). https://doi.org/10.1007/s10469-018-9475-8

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  • DOI: https://doi.org/10.1007/s10469-018-9475-8

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