{
  "id": "1508.01870",
  "version": "v1",
  "published": "2015-08-08T08:36:02.000Z",
  "updated": "2015-08-08T08:36:02.000Z",
  "title": "Invariable generation of the symmetric group",
  "authors": [
    "Sean Eberhard",
    "Kevin Ford",
    "Ben Green"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GR",
    "math.CO",
    "math.NT"
  ],
  "abstract": "We say that permutations $\\pi_1,\\dots, \\pi_r \\in \\mathcal{S}_n$ invariably generate $\\mathcal{S}_n$ if, no matter how one chooses conjugates $\\pi'_1,\\dots,\\pi'_r$ of these permutations, $\\pi'_1,\\dots,\\pi'_r$ generate $\\mathcal{S}_n$. We show that if $\\pi_1,\\pi_2,\\pi_3$ are chosen randomly from $\\mathcal{S}_n$ then, almost surely as $n \\rightarrow \\infty$, they do not invariably generate $\\mathcal{S}_n$. By contrast it was shown recently by Pemantle, Peres and Rivin that four random elements do invariably generate $\\mathcal{S}_n$ with positive probability. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-08T08:36:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric group",
      "invariable generation",
      "invariably generate",
      "permutations",
      "chooses conjugates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150801870E"
    }
  }
}