{
  "id": "2402.13247",
  "version": "v3",
  "published": "2024-02-20T18:57:54.000Z",
  "updated": "2024-06-07T21:13:26.000Z",
  "title": "On a bijection between a finite group to a non-cyclic group with divisibility of element orders",
  "authors": [
    "Mohsen Amiri"
  ],
  "comment": "The initial version provided was not the most recent iteration",
  "categories": [
    "math.GR"
  ],
  "abstract": "Consider a finite group $G$ of order $n$ with a prime divisor $p$. In this article, we establish, among other results, that if the Sylow $p$-subgroup of $G$ is neither cyclic nor generalized quaternion, then there exists a bijection $f$ from $G$ onto the abelian group $C_{\\frac{n}{p}}\\times C_p$ such that for every element $x$ in $G$, the order of $x$ divides the order of $f(x)$. This resolves Question 1.5 posed in [15]. As application of our results, we show that the group with the third largest value of the sum of element orders in the set of all finite groups of order $n$ is a solvable $p$-nilpotent group where $p$ is the smallest prime divisor of $n$ such that the Sylow $p$-subgroups are not cyclic.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-06-07T21:13:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite group",
      "element orders",
      "non-cyclic group",
      "divisibility",
      "smallest prime divisor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}