On maximal Abelian subgroups in finite 2-groups. (Q2471030)

A classification of nonabelian, finite \(2\)-groups \(G\) in which any two distinct maximal Abelian subgroups have cyclic intersection is given; it turns out that in groups with this property any two distinct maximal Abelian subgroups intersect in the centre \(Z(G)\). Groups with the latter property are then investigated, and as a result a problem of Heineken and Mann, about nonabelian \(2\)-groups \(G\) such that \(A/Z(G)\) is cyclic for all maximal Abelian subgroups \(A\), is solved.