{
  "id": "1911.04516",
  "version": "v1",
  "published": "2019-11-11T19:03:16.000Z",
  "updated": "2019-11-11T19:03:16.000Z",
  "title": "Boolean lattices in finite alternating and symmetric groups",
  "authors": [
    "Andrea Lucchini",
    "Mariapia Moscatiello",
    "Sebastien Palcoux",
    "Pablo Spiga"
  ],
  "comment": "25 pages, classification of Boolean lattices in symmetric and alternating groups",
  "categories": [
    "math.GR"
  ],
  "abstract": "Given a group $G$ and a subgroup $H$, we let $\\mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $\\mathcal{O}_{G}(H)$ is Boolean of rank at least $3$, when $G$ is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilizers of chains of regular partitions, and the other type arises by taking stabilizers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices, related to the dual Ore's theorem and to a problem of Kenneth Brown.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-11T19:03:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B25"
    ],
    "keywords": [
      "symmetric group",
      "finite alternating",
      "boolean lattices",
      "type arises",
      "dual ores theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}