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

Qinhai Zhang; Libo Zhao; Miaomiao Li; Yiqun Shen

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

Group Theory 2014-10-24 v1

Authors: Qinhai Zhang , Libo Zhao , Miaomiao Li , Yiqun Shen

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.

Keywords

finite p-groups abelian subgroups index p^3 A_t-groups classification Berkovich-Janko problem

Cite

@article{arxiv.1410.6226,
  title  = {Finite $p$-groups all of whose subgroups of index $p^3$ are abelian},
  author = {Qinhai Zhang and Libo Zhao and Miaomiao Li and Yiqun Shen},
  journal= {arXiv preprint arXiv:1410.6226},
  year   = {2014}
}

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

Abstract

Keywords

Cite

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