{
  "id": "2107.06248",
  "version": "v1",
  "published": "2021-07-13T17:11:06.000Z",
  "updated": "2021-07-13T17:11:06.000Z",
  "title": "Abelian sections of the symmetric groups with respect to their index",
  "authors": [
    "Luca Sabatini"
  ],
  "comment": "6 pages",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We show the existence of an absolute constant $\\alpha>0$ such that, for every $k \\geq 3$, $G:= \\mathop{\\mathrm{Sym}}(k)$, and for every $H \\leqslant G$ of index at least $3$, one has $|H/H'| \\leq \\exp \\left( \\frac{\\alpha \\cdot \\log|G:H|}{\\log \\log |G:H|} \\right)$. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-13T17:11:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B30",
      "20B35",
      "20F69"
    ],
    "keywords": [
      "symmetric groups",
      "abelian sections",
      "arbitrarily large finite groups",
      "absolute constant",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}