{
  "id": "2304.14200",
  "version": "v1",
  "published": "2023-04-27T14:05:31.000Z",
  "updated": "2023-04-27T14:05:31.000Z",
  "title": "On the maximum number of subgroups of a finite group",
  "authors": [
    "Marco Fusari",
    "Pablo Spiga"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Given a finite group $R$, we let $\\mathrm{Sub}(R)$ denote the collection of all subgroups of $R$. We show that $|\\mathrm{Sub}(R)|< c\\cdot |R|^{\\frac{\\log_2|R|}{4}}$, where $c<7.372$ is an explicit absolute constant. This result is asymptotically best possible. Indeed, as $|R|$ tends to infinity and $R$ is an elementary abelian $2$-group, the ratio $$\\frac{|\\mathrm{Sub}(R)|}{|R|^{\\frac{\\log_2|R|}{4}}}$$ tends to $c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-27T14:05:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite group",
      "maximum number",
      "explicit absolute constant",
      "elementary abelian",
      "collection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}