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When is the search of relatively maximal subgroups reduced to quotients? Full article

Journal Izvestiya Mathematics
ISSN: 1064-5632
Output data Year: 2022, Volume: 86, Number: 6, Pages: 1102–1122 Pages count : 20 DOI: 10.4213/im9277e
Tags finite group, complete class, X -maximal subgroup, Hall subgroup, reduction X -theorem.
Authors Го В. 4,5 , Revin D.O. 1,2,3
Affiliations
1 Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Sciences, Novosibirsk
2 Novosibirsk State University
3 Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
4 School of Science, Hainan University, Haikou, Hainan, P. R. China
5 Department of Mathematics, University of Science and Technology of China, Hefei, P. R. China

Funding (2)

1 Sobolev Institute of Mathematics FWNF-2022-0002
2 Russian Foundation for Basic Research 20-51-00007

Abstract: Let X be a class of finite groups closed with respect to subgroups, homomorphic images and extensions. Denote by kX(G) the number of conjugacy classes X-maximal subgroups of a finite group G. The natural problem to describe up to conjugacy X-maximal subgroups of a given finite group is complicated by the fact that it is not inductive. In particular, generally speaking, the image of an X-maximal subgroup is not X-maximal in the image of a homomorphism. Nevertheless, there are group homomorphisms which preserve the number of conjugacy classes of X-maximal subgroups (for example, homomorphisms whose kernels are X-groups). With respect to such homomorphisms, the image of an X-maximasubgroup is always X-maximal and, moreover, there is a natural bijection between the conjugacy classes of X-maximal subgroups of the image and preimage. Such homomorphisms are completely described in the paper. More precisely, it is proved that, for a homomorphism ϕ of a group G, the equality kX(G)=kX(imϕ) is satisfied if and only if kX(kerϕ)=1, which in turn is equivalent to the fact that the composition factors of the kernel ϕ belong to an explicitly given list.
Cite: Го В. , Revin D.O.
When is the search of relatively maximal subgroups reduced to quotients?
Izvestiya Mathematics. 2022. V.86. N6. P.1102–1122. DOI: 10.4213/im9277e WOS Scopus РИНЦ OpenAlex
Original: Го В. , Ревин Д.О.
Когда поиск относительно максимальных подгрупп редуцируется к факторгруппам?
Известия Российской академии наук. Серия математическая. 2022. Т.86. №6. С.79-100. DOI: 10.4213/im9277 РИНЦ MathNet OpenAlex
Dates:
Submitted: Oct 29, 2021
Accepted: Jan 30, 2022
Published online: May 2, 2023
Identifiers:
Web of science: WOS:000992259900003
Scopus: 2-s2.0-85165393590
Elibrary: 59310369
OpenAlex: W4381306350
Citing: Пока нет цитирований
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