Groups with many intersections of maximal subgroups (Q1576433)

A subgroup \(H\) of a group \(G\) is said to be an \(\mathcal M\)-subgroup if it is the intersection of a (possibly empty) set of maximal subgroups of \(G\). Moreover, the group \(G\) is called an IM-group if all its subgroups are \(\mathcal M\)-subgroups. The structure of soluble IM-groups has been completely described by \textit{F. Menegazzo} [Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat., VIII. Ser. 48, 559-562 (1970; Zbl 0216.08802)]. In the article under review groups are studied for which the set of non-\(\mathcal M\)-subgroups is small in some sense. In particular, the authors prove that if \(G\) is a hyperabelian group satisfying the minimal condition on non-\(\mathcal M\)-subgroups and if \([G,G']\) is an \(\mathcal M\)-subgroup of \(G\), then either \(G\) is a Chernikov group or an IM-group. Moreover, an example is constructed to show that there exists a metabelian group \(G\) (which is not a Chernikov group) such that \([G,G']\) is the unique non-\(\mathcal M\)-subgroup of \(G\). Finally, the authors also characterize (in a similar way) hyperabelian groups with finitely many conjugacy classes of non-\(\mathcal M\)-subgroups, and those for which the set of \(\mathcal M\)-subgroups is dense in the partially ordered set consisting of all subgroups.