{
  "id": "1606.08017",
  "version": "v1",
  "published": "2016-06-26T09:46:40.000Z",
  "updated": "2016-06-26T09:46:40.000Z",
  "title": "Projective linear groups as automorphism groups of chiral polytopes",
  "authors": [
    "Jérémie Moerenhout",
    "Dimitri Leemans",
    "Eugenia O'Reilly-Regueiro"
  ],
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "It is already known that the automorphism group of a chiral polyhedron is never isomorphic to $PSL(2,q)$ or $PGL(2,q)$ for any prime power $q$. In this paper, we show that $PSL(2,q)$ and $PGL(2,q)$ are never automorphism groups of chiral polytopes of rank at least $5$. Moreover, we show that $PGL(2,q)$ is the automorphism group of at least one chiral polytope of rank $4$ for every $q\\geq5$. Finally, we determine for which values of $q$ the group $PSL(2,q)$ is the automorphism group of a chiral polytope of rank $4$, except when $q=p^d\\equiv3\\pmod{4}$ where $d>1$ is not a prime power, in which case the problem remains unsolved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-26T09:46:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B11",
      "20G40"
    ],
    "keywords": [
      "automorphism group",
      "chiral polytope",
      "projective linear groups",
      "prime power",
      "problem remains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}