{
  "id": "2209.13037",
  "version": "v1",
  "published": "2022-09-26T21:34:29.000Z",
  "updated": "2022-09-26T21:34:29.000Z",
  "title": "Powers of commutators in linear algebraic groups",
  "authors": [
    "Benjamin Martin"
  ],
  "comment": "6 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let ${\\mathcal G}$ be a linear algebraic group over $k$, where $k$ is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let $G= {\\mathcal G}(k)$. We prove that if $\\gamma, \\delta\\in G$ such that $\\gamma$ is a commutator and $\\langle \\delta\\rangle= \\langle \\gamma\\rangle$ then $\\delta$ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz Principle from first-order model theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-26T21:34:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G15",
      "20F12",
      "03C98"
    ],
    "keywords": [
      "linear algebraic group",
      "commutator",
      "nonarchimedean local field",
      "first-order model theory",
      "pseudo-finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}