Covering monolithic groups with proper subgroups

TL;DR

This paper investigates the minimal number of proper subgroups needed to cover a finite non-cyclic group and explores a conjecture linking subgroup coverings to the group's monolithic structure.

Contribution

It proposes a method to approach Lucchini and Detomi's conjecture by directly studying monolithic groups and their properties.

Findings

The paper clarifies the relationship between subgroup coverings and monolithic groups.

It suggests a new approach to prove the conjecture using properties of monolithic groups.

Abstract

Given a finite non-cyclic group $G$, call $\sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\sigma(G) < \sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.