Minimum $\mathcal{F}$-covers: the supersolvable and metabelian cases

TL;DR

This paper investigates the existence of minimal groups that cover a set of groups within certain classes, providing negative results for supersolvable and metabelian classes.

Contribution

It answers a posed question by showing that minimum covers do not always exist within supersolvable and metabelian groups.

Findings

No minimum $\\mathcal{F}$-cover exists for supersolvable groups.

No minimum $\mathcal{F}$-cover exists for metabelian groups.

Abstract

Given a set $\mathcal{F}$ of finite groups, it is said that a group $G$ is an $\mathcal{F}$-cover if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. Moreover, $G$ is a minimum $\mathcal{F}$-cover if there is no $\mathcal{F}$-cover whose order is less than $|G|$. In [Cameron P. J., et al., Minimal cover groups, J. Algebra 660 (2024)], the authors pose the following question: For which classes $\mathcal{X}$ of groups, closed under taking subgroups and direct products, is it true that, if $\mathcal{F}$ is a set of $\mathcal{X}$-groups, then there is a minimum $\mathcal{F}$-cover which is an $\mathcal{X}$-group? In this paper, we give a negative answer in two cases: $\mathcal{X}\in \{``supersolvable", ``metabelian"\}.$