Finite 𝑝-groups all of whose 𝓐2-subgroups are generated by two elements

Abstract

Abstract Assume G is a finite p-group. We prove that if all 𝒜 2 {\mathcal{A}_{2}} -subgroups of G are generated by two elements, then so are all non-abelian subgroups of G. By using this result, we classify the p-groups which have at least two 𝒜 1 {\mathcal{A}_{1}} -subgroups and in which the intersection of every pair of distinct 𝒜 1 {\mathcal{A}_{1}} -subgroups equals the intersection of all the 𝒜 1 {\mathcal{A}_{1}} -subgroups. It turns out that such p-groups are the finite p-groups all of whose non-abelian subgroups are generated by two elements, and the converse also almost holds.