Finite $p$-groups all of whose subgroups of index $p^3$ are abelian

Abstract

Suppose that $G$ is a finite $p$-group. If all subgroups of index $p^t$ of $G$ are abelian and at least one subgroup of index $p^{t-1}$ of $G$ is not abelian, then $G$ is called an ${\mathcal{A}}_t$-group. In this paper, some information about ${\mathcal{A}}_t$-groups are obtained and ${\mathcal{A}}_3$-groups are completely classified. This solves an {\it old problem} proposed by Berkovich and Janko in their book. Abundant information about ${\mathcal{A}}_3$-groups are given.