{
  "id": "1905.00815",
  "version": "v1",
  "published": "2019-05-02T15:43:14.000Z",
  "updated": "2019-05-02T15:43:14.000Z",
  "title": "On Two Conjectures about the Sum of Element Orders",
  "authors": [
    "Morteza Baniasad Azad",
    "Behrooz Khosravi"
  ],
  "comment": "8 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a finite group and $\\psi(G) = \\sum_{g \\in G} o(g)$, where $o(g)$ denotes the order of $g \\in G$. First, we prove that if $G$ is a group of order $n$ and $\\psi(G) >31\\psi(C_n)/77$, where $C_n$ is the cyclic group of order $n$, then $G$ is supersolvable. This proves a conjecture of M.~{T\\u{a}rn\\u{a}uceanu}. Moreover, M. Herzog, P. Longobardi and M. Maj put forward the following conjecture: If $H\\leq G$, then $\\psi(G) \\leqslant \\psi(H) |G:H|^2$. In the sequel, by an example we show that this conjecture is not satisfied in general.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-02T15:43:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D60",
      "20F16"
    ],
    "keywords": [
      "element orders",
      "conjecture",
      "finite group",
      "cyclic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}