{
  "id": "1105.6066",
  "version": "v1",
  "published": "2011-05-30T18:49:36.000Z",
  "updated": "2011-05-30T18:49:36.000Z",
  "title": "On the divisibility of $#\\Hom(Γ,G)$ by $|G|",
  "authors": [
    "Fernando Rodriguez Villegas",
    "Cameron Gordon"
  ],
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $\\Gamma$ is a finitely generated group, then $|G|$ divides $#\\Hom(\\Gamma,G)$ for every finite group $G$ if and only if $\\Gamma$ has infinite abelianization. As a consequence we obtain some arithmetic properties of the number of subgroups of a given index in such a group $\\Gamma$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-30T18:49:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divisibility",
      "finite group",
      "infinite abelianization",
      "arithmetic properties",
      "finitely generated group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.6066R"
    }
  }
}