{
  "id": "2203.02362",
  "version": "v1",
  "published": "2022-03-04T14:55:50.000Z",
  "updated": "2022-03-04T14:55:50.000Z",
  "title": "Classification of non-solvable groups whose power graph is a cograph",
  "authors": [
    "Jendrik Brachter",
    "Eda Kaja"
  ],
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Cameron, Manna and Mehatari investigated the question of which finite groups admit a power graph that is a cograph, also called power-cograph groups (Journal of Algebra 591 (2022)). They give a classification for nilpotent groups and partial results for general groups. However, the authors point out number theoretic obstacles towards a classification. These arise when the groups are assumed to be isomorphic to PSL 2 (q) or Sz(q) and are likely to be hard. In this paper, we prove that these number theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 (q) or Sz(q). For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg-Kegel graph is connected. Moreover, we reduce the case where the Gruenberg-Kegel graph is disconnected to the classification of p-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-04T14:55:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E34",
      "05C25"
    ],
    "keywords": [
      "classification",
      "power graph",
      "non-solvable groups",
      "solvable power-cograph groups",
      "gruenberg-kegel graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}