Covering monolithic groups with proper subgroups

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.

I wrote this paper for the Proceedings of the conference "Ischia Group Theory 2012" (March, 26th - 29th 2012)