{
  "id": "1807.04959",
  "version": "v1",
  "published": "2018-07-13T07:56:21.000Z",
  "updated": "2018-07-13T07:56:21.000Z",
  "title": "A note on some special $p$-groups",
  "authors": [
    "Farangis Johari",
    "Peyman Niroomand"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Recently Rai obtained an upper bound for the order of the Schur multiplier of a $d$-generator special $p$-group when its derived subgroup has the maximum value $ p^{\\frac{1}{2}d(d-1)}$ for $ d\\geq 3 $ and $ p\\neq 2. $ Here we try to obtain the Schur multiplier, the exterior square and the tensor square of such $p$-groups. Then we specify which ones are capable. Moreover, we give an upper bound for the order of the Schur multiplier, the exterior product and the tensor square of a $d$-generator special $p$-group $ G $ when $ |G'|=p^{\\frac{1}{2}d(d-1)-1}$ for $ d\\geq 3 $ and $ p\\neq 2. $ Additionally, when $ G $ is of exponent $ p, $ we give the structure of $ G. $",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-13T07:56:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15",
      "20E34"
    ],
    "keywords": [
      "schur multiplier",
      "generator special",
      "upper bound",
      "tensor square",
      "exterior product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}