{
  "id": "2401.05937",
  "version": "v1",
  "published": "2024-01-11T14:23:25.000Z",
  "updated": "2024-01-11T14:23:25.000Z",
  "title": "On the lattice of closed subgroups of a profinite group",
  "authors": [
    "Francesco de Giovanni",
    "Iker de las Heras",
    "Marco Trombetti"
  ],
  "comment": "23 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behaviour of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-11T14:23:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "closed subgroups",
      "open subgroups",
      "dedekind modular law",
      "studying profinite groups",
      "results concerning decomposability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}