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. 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.

Keywords

$p$-groups')">Finite $p$-groups / $p$-groups')">Minimal non-abelian $p$-groups / ${\mathcal {A}}_t$-groups')">${\mathcal {A}}_t$-groups

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