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

TL;DR

The paper investigates for which classes $\mathcal{X}$ closed under subgroups and direct products there exists a minimum $\mathcal{F}$-cover that lies in $\mathcal{X}$. Building on known results—such as the nilpotent case admitting a minimum cover within the class and the cyclic-case yielding a cyclic minimum cover—the authors provide negative answers for $\mathcal{X}$ in {supersolvable, metabelian}. They construct infinite families of $\mathcal{F}$ formed from $\mathcal{X}$-groups whose unique minimum $\mathcal{F}$-cover lies outside the class, giving concrete instances: for supersolvable, the unique minimum cover is $C_p\times(C_3^2\rtimes C_4)$; for metabelian, it is $C_p\times A_5$. The proofs leverage Sylow theory, Schur-Zassenhaus, and quotient-structure analyses to tightly constrain possibilities and show that the minimum covers escape the targeted classes. Overall, the results delineate limits on preserving the defining properties of $\mathcal{X}$ in minimum $\mathcal{F}$-covers, highlighting nuanced behavior beyond nilpotent and cyclic settings.

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: $\mathcal{X}\in \{``supersolvable", ``metabelian"\}.$