{
  "id": "1801.09928",
  "version": "v1",
  "published": "2018-01-30T10:56:05.000Z",
  "updated": "2018-01-30T10:56:05.000Z",
  "title": "Invariable generation of permutation and linear groups",
  "authors": [
    "Gareth M. Tracey"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "A subset $\\left\\{x_{1},x_{2},\\hdots,x_{d}\\right\\}$ of a group $G$ \\emph{invariably generates} $G$ if $\\left\\{x_{1}^{g_{1}},x_{2}^{g_{2}},\\hdots,x_{d}^{g_{d}}\\right\\}$ generates $G$ for every $d$-tuple $(g_{1},g_{2}\\hdots,g_{d})\\in G^{d}$. We prove that a finite completely reducible linear group of dimension $n$ can be invariably generated by $\\left\\lfloor \\frac{3n}{2}\\right\\rfloor$ elements. We also prove tighter bounds when the field in question has order $2$ or $3$. Finally, we prove that a transitive [respectively primitive] permutation group of degree $n\\geq 2$ [resp. $n\\geq 3$] can be invariably generated by $O\\left(\\frac{n}{\\sqrt{\\log{n}}}\\right)$ [resp. $O\\left(\\frac{\\log{n}}{\\sqrt{\\log{\\log{n}}}}\\right)$] elements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-30T10:56:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariable generation",
      "tighter bounds",
      "permutation group",
      "reducible linear group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}