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

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 $\varphi$ from a group $G$, the equality ${\mathrm{k}}_{\mathfrak{X}}(G)={\mathrm{k}}_{\mathfrak{X}}(\mathrm{im}\varphi )$ holds if and only if ${\mathrm{k}}_{\mathfrak{X}}(\ker \varphi )=1$, which in turn is equivalent to the fact that the composition factors of the kernel of $\varphi$ belong to an explicitly given list.

in Russian

References in corpus (2)

• On the number of classes of conjugate Hall subgroups in finite simple groups
• The reduction theorem for relatively maximal subgroups