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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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