{
  "id": "1903.06245",
  "version": "v1",
  "published": "2019-03-14T20:51:46.000Z",
  "updated": "2019-03-14T20:51:46.000Z",
  "title": "Commutators in finite p-groups with 3-generator derived subgroup",
  "authors": [
    "Iker de las Heras"
  ],
  "comment": "15 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "It is well known that, in general, the set of commutators of a group $G$ may not be a subgroup. Guralnick showed that if $G$ is a finite $p$-group with $p\\ge 5$ such that $G'$ is abelian and $3$-generator, then all the elements of the derived subgroup are commutators. In this paper, we extend Guralnick's result by showing that the condition of $G'$ to be abelian is not needed. In this way, we complete the study of this property in finite $p$-groups in terms of the number of generators of the derived subgroup. We will also see that the same result is true when the action of $G$ on $G'$ is uniserial modulo $(G')^p$ and $|G':(G')^p|$ does not exceed $p^{p-1}$. Finally, we will prove that analogous results are satisfied when working with pro-$p$ groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-14T20:51:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "derived subgroup",
      "finite p-groups",
      "commutators",
      "extend guralnicks result",
      "uniserial modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}