{
  "id": "2207.01215",
  "version": "v1",
  "published": "2022-07-04T05:55:43.000Z",
  "updated": "2022-07-04T05:55:43.000Z",
  "title": "Derangements in wreath products of permutation groups",
  "authors": [
    "Vishnuram Arumugam",
    "Heiko Dietrich",
    "S. P. Glasby"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Given a finite group $G$ acting on a set $X$ let $\\delta_k(G,X)$ denote the proportion of elements in $G$ that have exactly $k$ fixed points in $X$. Let $\\mathrm{S}_n$ denote the symmetric group acting on $[n]=\\{1,2,\\dots,n\\}$. For $A\\le\\mathrm{S}_m$ and $B\\le\\mathrm{S}_n$, the permutational wreath product $A\\wr B$ has two natural actions and we give formulas for both, $\\delta_k(A\\wr B,[m]{\\times}[n])$ and $\\delta_k(A\\wr B,[m]^{[n]})$. We prove that for $k=0$ the values of these proportions are dense in the intervals $[\\delta_0(B,[n]),1]$ and $[\\delta_0(A,[m]),1]$. Among further result, we provide estimates for $\\delta_0(G,[m]^{[n]})$ for subgroups $G\\leq \\mathrm{S}_m\\wr\\mathrm{S}_n$ containing $\\mathrm{A}_m^{[n]}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-04T05:55:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B07",
      "20B35",
      "05A05"
    ],
    "keywords": [
      "permutation groups",
      "derangements",
      "permutational wreath product",
      "natural actions",
      "proportion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}