On the intersection of $\mathfrak{F}$-maximal subgroups of a finite group

TL;DR

This paper studies the intersection of all $rak{F}$-maximal subgroups in finite groups, characterizing when this intersection behaves trivially, and connects to previous results on hereditary formations.

Contribution

It provides a new characterization of the triviality of the intersection of $rak{F}$-maximal subgroups in terms of the formation containing certain groups.

Findings

Proves $ ext{Int}_{rak{F}}(G/ ext{Int}_{rak{F}}(G)) o 1$ under specific conditions.

Characterizes when the intersection of $rak{F}$-maximal subgroups is trivial.

Recovers and extends previous results by Skiba and others.

Abstract

We investigate the properties of the intersection $\mathrm{Int}_{\mathfrak{F}}(G)$ of all $\mathfrak{F}$-maximal subgroups of a finite group $G$ for a hereditary formation $\mathfrak{F}$ of finite groups. We prove that $\mathrm{Int}_{\mathfrak{F}}(G/\mathrm{Int}_{\mathfrak{F}}(G))\simeq 1$ holds for any finite group $G$ if and only if $\mathfrak{F}$ contains every group $G$ all of whose $\mathfrak{F}$-subgroups are $\mathfrak{F}$-subnormal. As corollaries we obtain the results of A. N. Skiba (2011), J. C. Beidleman and H. Heineken (2011) about $\mathrm{Int}_{\mathfrak{F}}(G)$ for a hereditary saturated formation $\mathfrak{F}$.