Given a finite non-cyclic group G, call σ(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 σ(G)<σ(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.Comment: I wrote this paper for the Proceedings of the conference "Ischia
Group Theory 2012" (March, 26th - 29th 2012