{
  "id": "2101.12116",
  "version": "v1",
  "published": "2021-01-28T17:02:38.000Z",
  "updated": "2021-01-28T17:02:38.000Z",
  "title": "On the finite index subgroups of Houghton's groups",
  "authors": [
    "Charles Garnet Cox"
  ],
  "comment": "6 pages; comments welcome!",
  "categories": [
    "math.GR"
  ],
  "abstract": "Houghton's groups $H_2, H_3, \\ldots$ have been studied in many contexts, and various results exist for their finite index subgroups. In this note we describe all of the finite index subgroups of each Houghton group, and their isomorphism types. Using the standard notation that $d(G)$ denotes the minimal size of generating set for $G$ we then show, for each $n\\in \\{2, 3,\\ldots\\}$ and $U$ of finite index in $H_n$, that $d(U)\\in\\{d(H_n), d(H_n)+1\\}$ and characterise when each of these cases occurs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-28T17:02:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite index subgroups",
      "houghtons groups",
      "houghton group",
      "isomorphism types",
      "standard notation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}