Groups of prime power order covered by a certain number of proper subgroups (Q1734103)

A covering of a group \(G\) is a set of proper subgroups whose set-theoretic union is the whole of \(G\). A covering is irredundant if no proper subset is a cover itself. \textit{Y. Berkovich} asked in [Glas. Mat., III. Ser. 45, No. 2, 415--429 (2010; Zbl 1216.20011)] whether there are finite \(p\)-groups with an irredundant covering consisting of \(n\) subgroups, with \(p + 1 < n < 2 p\), and in case whether it is possible to classify them. In this ingenious short paper, the author shows that for \(p\) an odd prime, any group with minimum number of generators at least \(3\) has an irredundant covering consisting of \(3 (p + 1)/2\) subgroups. This shows that for \(p > 3\) there is no classification as above. The proof consists in constructing such a covering for the elementary abelian group of order \(p^{3}\), using the projective triangle in the projective plane.