COVERING MONOLITHIC GROUPS WITH PROPER SUBGROUPS

Given a finite non-cyclic group G, call (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 (G) < (G=N) for every non-trivial normal subgroup N of G then G is 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.