{
  "id": "2407.16003",
  "version": "v1",
  "published": "2024-07-22T19:18:21.000Z",
  "updated": "2024-07-22T19:18:21.000Z",
  "title": "Regular polytopes of rank $n/2$ for transitive groups of degree $n$",
  "authors": [
    "Maria Elisa Fernandes",
    "Claudio Alexandre Piedade"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Previous research established that the maximal rank of the abstract regular polytopes whose automorphism group is a transitive proper subgroup of $\\Sym_n$ is $n/2 + 1$, with only two polytopes attaining this rank, both of which having odd ranks. In this paper, we investigate the case where the rank is equal to $n/2$ ($n\\geq 14$). Our analysis reveals that reducing the rank by one results in a substantial increase in the number of regular polytopes ($33$ distinct families are discovered) covering all possible ranks (even and odd).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-22T19:18:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B11",
      "20B35",
      "20B30",
      "05C25"
    ],
    "keywords": [
      "transitive groups",
      "abstract regular polytopes",
      "maximal rank",
      "odd ranks",
      "transitive proper subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}