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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
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