{
  "id": "2003.06384",
  "version": "v1",
  "published": "2020-03-13T17:07:25.000Z",
  "updated": "2020-03-13T17:07:25.000Z",
  "title": "An approach to Quillen's conjecture via centralizers of simple groups",
  "authors": [
    "Kevin Ivan Piterman"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GR",
    "math.AT"
  ],
  "abstract": "We show that, for any given subgroup $H$ of a finite group $G$, the Quillen poset $\\mathcal{A}_p(G)$ of nontrivial elementary abelian $p$-subgroups, is obtained from $\\mathcal{A}_p(H)$ by attaching elements via their centralizers in $H$. We use this idea to study Quillen's conjecture, which asserts that if $\\mathcal{A}_p(G)$ is contractible then $G$ has a nontrivial normal $p$-subgroup. We prove that the original conjecture is equivalent to the $\\mathbb{Z}$-acyclic version of the conjecture (obtained by replacing contractible by $\\mathbb{Z}$-acyclic). We also work with the $\\mathbb{Q}$-acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of order divisible by $p$ and $p$-rank at least $2$. This allows to extend results of Aschbacher-Smith and to establish the strong conjecture for groups of $p$-rank at most $4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-13T17:07:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J05",
      "20D05",
      "20D25",
      "20D30",
      "05E18",
      "06A11"
    ],
    "keywords": [
      "simple groups",
      "centralizers",
      "nontrivial elementary abelian",
      "study quillens conjecture",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}