When is the search of relatively maximal subgroups reduced to quotient groups?
- Авторлар: Guo W.B.1,2, Revin D.O.3,4,5
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Мекемелер:
- School of Science, Hainan University
- University of Science and Technology of China
- Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
- N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
- Novosibirsk State University
- Шығарылым: Том 86, № 6 (2022)
- Беттер: 79-100
- Бөлім: Articles
- URL: https://ogarev-online.ru/1607-0046/article/view/133890
- DOI: https://doi.org/10.4213/im9277
- ID: 133890
Дәйексөз келтіру
Ашық рұқсат
Рұқсат берілді
Тек жазылушылар үшін
Аннотация
Let $\mathfrak{X}$ be a class finite groups closed under taking subgroups, homomorphic images, and extensions, andlet $\mathrm{k}_{\mathfrak{X}}(G)$ be the number of conjugacy classes $\mathfrak{X}$-maximal subgroups of a finite group $G$.The natural problem calling for a description, up to conjugacy, ofthe $\mathfrak{X}$-maximal subgroups of a given finite group is not inductive.In particular, generally speaking, the image of an $\mathfrak{X}$-maximalsubgroup is not $\mathfrak{X}$-maximal in the image of a homomorphism.Nevertheless, there exist group homomorphisms that preserve the number of conjugacy classes of maximal$\mathfrak{X}$-subgroups (for example, the homomorphisms whose kernels are $\mathfrak{X}$-groups).Under such homomorphisms, the image of an $\mathfrak{X}$-maximal subgroup is always $\mathfrak{X}$-maximal,and, moreover, there is a natural bijection between the conjugacy classesof $\mathfrak{X}$-maximal subgroups of the image and preimage.In the present paper, all such homomorphisms arecompletely described.More precisely, it is shown that, for a homomorphism $\phi$from a group $G$, the equality $\mathrm{k}_{\mathfrak{X}}(G)=\mathrm{k}_{\mathfrak{X}}(\operatorname{im} \phi)$holds if and only if $\mathrm{k}_{\mathfrak{X}}(\ker \phi)=1$,which in turn is equivalent to the fact that the composition factors of the kernel of $\phi$ lie in an explicitly given list.
Негізгі сөздер
Авторлар туралы
Wen Guo
School of Science, Hainan University; University of Science and Technology of China
Email: wguo@ustc.edu.cn
Doctor of physico-mathematical sciences
Danila Revin
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences; N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences; Novosibirsk State University
Email: revin@math.nsc.ru
Doctor of physico-mathematical sciences, no status
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