{
  "id": "1311.6148",
  "version": "v2",
  "published": "2013-11-24T17:49:27.000Z",
  "updated": "2014-02-11T22:37:31.000Z",
  "title": "On finite groups in which coprime commutators are covered by few cyclic subgroups",
  "authors": [
    "Cristina Acciarri",
    "Pavel Shumyatsky"
  ],
  "comment": "Final version, referee's suggestions added",
  "categories": [
    "math.GR"
  ],
  "abstract": "The coprime commutators $\\gamma_j^*$ and $\\delta_j^*$ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. They are defined as follows. Let $G$ be a finite group. Every element of $G$ is both a $\\gamma_1^*$-commutator and a $\\delta_0^*$-commutator. Now let $j\\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $\\gamma_{j-1}^*$-commutators. An element $g$ is a $\\gamma_j^*$-commutator if there exist $a\\in X$ and $b\\in G$ such that $g=[a,b]$ and $(|a|,|b|)=1$. For $j\\geq 1$ let $Y$ be the set of all elements of $G$ that are powers of $\\delta_{j-1}^*$-commutators. The element $g$ is a $\\delta_j^*$-commutator if there exist $a,b\\in Y$ such that $g=[a,b]$ and $(|a|,|b|)=1$. The subgroups of $G$ generated by all $\\gamma_j^*$-commutators and all $\\delta_j^*$-commutators are denoted by $\\gamma_j^*(G)$ and $\\delta_j^*(G)$, respectively. For every $j\\geq2$ the subgroup $\\gamma_j^*(G)$ is precisely the last term of the lower central series of $G$ (which throughout the paper is denoted by $\\gamma_\\infty(G)$) while for every $j\\geq1$ the subgroup $\\delta_j^*(G)$ is precisely the last term of the lower central series of $\\delta_{j-1}^*(G)$, that is, $\\delta_j^*(G)=\\gamma_\\infty(\\delta_{j-1}^*(G))$. In the present paper we prove that if $G$ possesses $m$ cyclic subgroups whose union contains all $\\gamma_j^*$-commutators of $G$, then $\\gamma_j^*(G)$ contains a subgroup $\\Delta$, of $m$-bounded order, which is normal in $G$ and has the property that $\\gamma_{j}^{*}(G)/\\Delta$ is cyclic. If $j\\geq2$ and $G$ possesses $m$ cyclic subgroups whose union contains all $\\delta_j^*$-commutators of $G$, then the order of $\\delta_j^*(G)$ is $m$-bounded.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-11T22:37:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F14",
      "20D25"
    ],
    "keywords": [
      "finite group",
      "cyclic subgroups",
      "coprime commutators",
      "lower central series",
      "union contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.6148A"
    }
  }
}