Finite groups which have irreduntant covers of big cardinality (Q6635331)

A cover of a group \(G\) is a collection of proper subgroups whose union is \(G\). The cover is irredundant if no proper subcollection is also a cover and \(\sigma(G)\) is the smallest integer \(n\) such that the group \(G\) is covered by \(n\) proper subgroups.\N\NIt is well-known that there are no finite groups with \(\sigma(G)=2\). Moreover, \textit{G. Scorza} [Bollettino U.M.I. 5, 216--218 (1926; JFM 52.0113.03)] classified the groups with \(\sigma(G)=3\). \textit{D. Greco} [Rend. Accad. Sci. Fis. Mat., IV. Ser., Napoli 23, 49--59 (1957; Zbl 0166.28304)] has dealt with the cases \(\sigma(G)=4\) and \(\sigma(G)=5\). More information on groups with small values of \(\sigma\) can be found in [\textit{J. H. E. Cohn}, Math. Scand. 75, No. 1, 44--58 (1994; Zbl 0833.20028); \textit{M. J. Tomkinson}, Math. Scand. 81, No. 2, 191--198 (1997; Zbl 0905.20014)].\N\NIn the paper under review, the authors consider a dual problem. Let \(\lambda(G)\) be the maximum number of subgroups in an irredundant covering of group \(G\) (see [\textit{J. R. Rogério}, Commun. Algebra 42, No. 10, 4498--4508 (2014; Zbl 1304.20043)]). They prove that the finite groups with \(\lambda(G)=|G|-t\), where \(t \leq 5\), are solvable and classify such groups.