{
  "id": "2501.02249",
  "version": "v1",
  "published": "2025-01-04T10:27:47.000Z",
  "updated": "2025-01-04T10:27:47.000Z",
  "title": "Extensions of a theorem of P. Hall on indexes of maximal subgroups",
  "authors": [
    "Antonio Beltrán",
    "Changguo Shao"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "We extend a classical theorem of P. Hall that claims that if the index of every maximal subgroup of a finite group $G$ is a prime or the square of a prime, then $G$ is solvable. Precisely, we prove that if one allows, in addition, the possibility that every maximal subgroup of $G$ is nilpotent instead of having prime or squared-prime index, then $G$ continues to be solvable. Likewise, we obtain the solvability of $G$ when we assume that every proper non-maximal subgroup of $G$ lies in some subgroup of index prime or squared prime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-04T10:27:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E28",
      "20D15",
      "20D06"
    ],
    "keywords": [
      "extensions",
      "proper non-maximal subgroup",
      "finite group",
      "squared-prime index",
      "index prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}