When is the search of relatively maximal subgroups reduced to quotient groups?

Author:

Guo Wen Bin12,Revin Danila Olegovich345

Affiliation:

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

2. School of Science, Hainan University, Haikou, Hainan, P. R. China

3. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

4. N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

5. Novosibirsk State University

Abstract

Let $\mathfrak{X}$ be a class finite groups closed under taking subgroups, homomorphic images, and extensions, and let $\mathrm{k}_{\mathfrak{X}}(G)$ 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, of the $\mathfrak{X}$-maximal subgroups of a given finite group is not inductive. In particular, generally speaking, the image of an $\mathfrak{X}$-maximal subgroup 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 classes of $\mathfrak{X}$-maximal subgroups of the image and preimage. In the present paper, all such homomorphisms are completely 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.

Funder

Russian Foundation for Basic Research

National Natural Science Foundation of China

Ministry of Science and Higher Education of the Russian Federation

Publisher

Steklov Mathematical Institute

Subject

General Mathematics

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