{
  "id": "1410.6226",
  "version": "v1",
  "published": "2014-10-23T02:10:55.000Z",
  "updated": "2014-10-23T02:10:55.000Z",
  "title": "Finite $p$-groups all of whose subgroups of index $p^3$ are abelian",
  "authors": [
    "Qinhai Zhang",
    "Libo Zhao",
    "Miaomiao Li",
    "Yiqun Shen"
  ],
  "categories": [
    "math.GR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-23T02:10:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15"
    ],
    "keywords": [
      "old problem",
      "abundant information"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.6226Z"
    }
  }
}