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.

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

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