{
  "id": "1311.2983",
  "version": "v2",
  "published": "2013-11-12T23:54:02.000Z",
  "updated": "2014-03-31T22:47:51.000Z",
  "title": "A group sum inequality and its application to power graphs",
  "authors": [
    "Brian Curtin",
    "Gholam Reza Pourgholi"
  ],
  "doi": "10.1017/S0004972714000434",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Let $G$ be a finite group of order $n$, and let $C_n$ be the cyclic group of order $n$. We show that $\\sum_{g \\in C_n} \\phi(\\mathrm{o}(g))\\geq \\sum_{g \\in G} \\phi(\\mathrm{o}(g))$, with equality if and only if $G$ is isomorphic to $C_n$. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of undirected edges in its directed power graph.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-31T22:47:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "20F99"
    ],
    "keywords": [
      "group sum inequality",
      "application",
      "finite group",
      "cyclic group",
      "maximum number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.2983C"
    }
  }
}