{
  "id": "2106.01219",
  "version": "v1",
  "published": "2021-06-02T15:10:29.000Z",
  "updated": "2021-06-02T15:10:29.000Z",
  "title": "Base sizes of primitive permutation groups",
  "authors": [
    "Mariapia Moscatiello",
    "Colva M. Roney-Dougal"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let G be a permutation group, acting on a set \\Omega of size n. A subset B of \\Omega is a base for G if the pointwise stabilizer G_(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r \\geq 1 such that Alt(m)^r \\unlhd G \\leq Sym(m) \\wr Sym(r), where the action of Sym(m) is on k-element subsets of {1,...,m} and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M24 in its natural action on 24 points, or b(G) \\leq \\lceil \\log n\\rceil+1. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b(G) > log n + 1, so our bound is optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-02T15:10:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B15",
      "20B10"
    ],
    "keywords": [
      "primitive permutation groups",
      "base sizes",
      "large base",
      "mathieu group m24",
      "wreath product acts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}