{
  "id": "1907.08477",
  "version": "v1",
  "published": "2019-07-19T12:12:46.000Z",
  "updated": "2019-07-19T12:12:46.000Z",
  "title": "A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group",
  "authors": [
    "Andrea Lucchini",
    "Mariapia Moscatiello",
    "Pablo Spiga"
  ],
  "comment": "8 pages, we answer a question of Peter Cameron on maximal systems of imprimitivity, see https://cameroncounts.wordpress.com/2016/11/28/road-closures-and-idempotent-generated-semigroups/",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We show that, there exists a constant $a$ such that, for every subgroup $H$ of a finite group $G$, the number of maximal subgroups of $G$ containing $H$ is bounded above by $a|G:H|^{3/2}$. In particular, a transitive permutation group of degree $n$ has at most $an^{3/2}$ maximal systems of imprimitivity. When $G$ is soluble, generalizing a classic result of Tim Wall, we prove a much stroger bound, that is, the number of maximal subgroups of $G$ containing $H$ is at most $|G:H|-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-19T12:12:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite transitive permutation group",
      "maximal systems",
      "polynomial bound",
      "imprimitivity",
      "maximal subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}