{
  "id": "2505.06130",
  "version": "v1",
  "published": "2025-05-09T15:36:17.000Z",
  "updated": "2025-05-09T15:36:17.000Z",
  "title": "Powers of commutators in infinite groups",
  "authors": [
    "Daan Heus"
  ],
  "comment": "17 pages; 1 figure; this paper is based on my bachelor's and master's theses, written at Leiden University under the supervision of Hendrik Lenstra",
  "categories": [
    "math.GR"
  ],
  "abstract": "Given elements $x,u,z$ in a finite group $G$ such that $z$ is the commutator of $x$ and $u$, and the orders of $x$ and $z$ divide respectively integers $k,m \\geq 2$, and given an integer $r$ that is coprime to $k$ and $m$, there exists $w \\in G$ such that the commutator of $x^r$ and $w$ is conjugate to $z^r$. If instead we are given elements $x,y,z \\in G$ such that $xy = z$, whose respective orders divide integers $k,l,m \\geq 2$, and are given an integer $r$ that is coprime to $k,l$ and $m$, then there exist $x'$, $y'$ and $z'$ conjugate to respectively $x^r$, $y^r$ and $z^r$ such that $x'y' = z'$. In this paper we completely answer the natural question for which values of $k,l,m,r$ every group has these properties. The proof uses combinatorial group theory and properties of the projective special linear group $\\mathrm{PSL}_2(\\mathbb{R})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-09T15:36:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F12",
      "20E45"
    ],
    "keywords": [
      "infinite groups",
      "commutator",
      "projective special linear group",
      "respective orders divide integers",
      "combinatorial group theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}