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

TL;DR

This paper investigates conditions under which the problem of classifying maximal subgroups in finite groups can be simplified by analyzing quotients, focusing on homomorphisms that preserve the number of conjugacy classes of such subgroups.

Contribution

It characterizes homomorphisms that preserve the number of conjugacy classes of maximal subgroups and relates this to the composition factors of their kernels.

Findings

Homomorphisms with kernels in a specific list preserve the number of conjugacy classes.

The equality of the number of maximal subgroups in a group and its image holds iff the kernel's composition factors are from a known list.

The paper provides a complete description of homomorphisms that maintain the maximal subgroup structure.

Abstract

Let ${\mathfrak{X}}$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Denote by ${\mathrm{k}}_{\mathfrak{X}}(G)$ the number of conjugacy classes ${\mathfrak{X}}$-maximal subgroups of a finite group $G$. The natural problem to describe up to conjugacy ${\mathfrak{X}}$-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 ${\mathfrak{X}}$-maximal subgroup is not ${\mathfrak{X}}$-maximal in the image of a homomorphism. Nevertheless, there are group homomorphisms which preserve the number of conjugacy classes of ${\mathfrak{X}}$-maximal 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. All such homomorphisms are completely described in the paper. More precisely, it is proved that, for a homomorphism $\phi$ from a group $G$, the equality ${\mathrm{k}}_{\mathfrak{X}}(G)={\mathrm{k}}_{\mathfrak{X}}(\mathrm{im}\, \phi)$ 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 $\phi$ belong to an explicitly given list.