Finite p-Groups All of Whose Subgroups of Index p2 Are Abelian

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. We use \({\mathcal {A}}_0\)-group to denote an abelian group. From the definition, we know every finite non-abelian \(p\)-group can be regarded as an \({\mathcal {A}}_t\)-group for some positive integer \(t\). \({\mathcal {A}}_1\)-groups and \({\mathcal {A}}_2\)-groups have been classified. Classifying \({\mathcal {A}}_3\)-groups is an old problem. In this paper, some general properties about \({\mathcal {A}}_t\)-groups are given. \({\mathcal {A}}_3\)-groups are completely classified up to isomorphism. Moreover, we determine the Frattini subgroup, the derived subgroup and the center of every \({\mathcal {A}}_3\)-group, and give the number of \({\mathcal {A}}_1\)-subgroups and the triple \((\mu _0,\mu _1,\mu _2)\) of every \({\mathcal {A}}_3\)-group, where \(\mu _i\) denotes the number of \({\mathcal {A}}_i\)-subgroups of index \(p\) of \({\mathcal {A}}_3\)-groups.