{
  "id": "2211.04044",
  "version": "v1",
  "published": "2022-11-08T06:59:39.000Z",
  "updated": "2022-11-08T06:59:39.000Z",
  "title": "Normalisers of Sylow subgroups in reflection groups",
  "authors": [
    "Kane Douglas Townsend"
  ],
  "comment": "12 pages, 1 figure, 2 tables",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $W$ be a finite reflection group, $\\ell$ a prime divisor of $|W|$ and $S_\\ell$ a Sylow $\\ell$-subgroup of $W$. It can be shown that there exists a unique parabolic subgroup $P$ minimally containing $S_\\ell$. We prove that there exists a simple system of $W$ such that $P$ is a standard parabolic subgroup and $N_W(S_\\ell)=N_{P}(S_\\ell) \\rtimes U,$ where $U$ is the setwise stabiliser of the simple roots of $P$. We also determine when an analogous decomposition of the normaliser of a Sylow $\\ell$-subgroup exists when we replace $P$ by a reflection subgroup minimally containing $S_\\ell$. Furthermore, it is shown that if $W$ is crystallographic and has no proper parabolic subgroup containing a Sylow $\\ell$-subgroup of $W$, then there is a Sylow $\\ell$-subgroup normalised by the Dynkin diagram automorphisms of $W$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-08T06:59:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "20D20"
    ],
    "keywords": [
      "sylow subgroups",
      "normaliser",
      "finite reflection group",
      "standard parabolic subgroup",
      "dynkin diagram automorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}