{
  "id": "1907.00513",
  "version": "v1",
  "published": "2019-07-01T02:06:06.000Z",
  "updated": "2019-07-01T02:06:06.000Z",
  "title": "Elementary Proof of a Theorem of Hawkes, Isaacs and Özaydin",
  "authors": [
    "Matthé van der Lee"
  ],
  "comment": "9 pages",
  "categories": [
    "math.GR",
    "math.CO",
    "math.NT"
  ],
  "abstract": "We present an elementary proof of the theorem of Hawkes, Isaacs and \\\"Ozaydin, which states that $\\Sigma\\,\\mu_{G}(H,K)\\equiv 0$ mod $d$, where $\\mu_{G}$ denotes the M\\\"obius function for the subgroup lattice of a finite group $G$, $H$ ranges over the conjugates of a given subgroup $F$ of $G$ with $[G:F]$ divisible by $d$, and $K$ over the supergroups of the $H$ for which $[K:H]$ divides $d$. We apply the theorem to obtain a result on the number of solutions of $|\\langle H,g\\rangle|\\mid n$, for said $H$ and a natural number $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-01T02:06:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D30",
      "11A25",
      "05E15"
    ],
    "keywords": [
      "elementary proof",
      "finite group",
      "natural number",
      "subgroup lattice",
      "supergroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}