{
  "id": "2501.13406",
  "version": "v1",
  "published": "2025-01-23T06:08:09.000Z",
  "updated": "2025-01-23T06:08:09.000Z",
  "title": "Finite groups admitting a regular tournament $m$-semiregular representation",
  "authors": [
    "Dein Wong",
    "Songnian Xu",
    "Chi Zhang",
    "Jinxing Zhao"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "For a positive integer $m$, a finite group $G$ is said to admit a tournament $m$-semiregular representation (TmSR for short) if there exists a tournament $\\Gamma$ such that the automorphism group of $\\Gamma$ is isomorphic to $G$ and acts semiregularly on the vertex set of $\\Gamma$ with $m$ orbits. Clearly, every finite group of even order does not admit a TmSR for any positive integer $m$, and T1SR is the well-known tournament regular representation (TRR for short). In 1986, Godsil \\cite{god} proved, by a probabilistic approach, that the only finite groups of odd order without a TRR are $\\mathbb{Z}_3^2$ and $\\mathbb{Z}_3^3$ . More recently, Du \\cite{du} proved that every finite group of odd order has a TmSR for every $m \\geq 2$. The author of \\cite{du} observed that a finite group of odd order has no regular TmSR when $m$ is an even integer, a group of order $1$ has no regular T3SR, and $\\mathbb{Z}_3^2$ admits a regular T3SR. At the end of \\cite{du}, Du proposed the following problem. \\noindent{\\sf\\it Problem.} \\ \\ {\\it For every odd integer $m\\geq 3$, classify finite groups of odd order which have a regular TmSR.} The motivation of this paper is to give an answer for the above problem. We proved that if $G$ is a finite group with odd order $n>1$, then $G$ admits a regular TmSR for any odd integer $m\\geq 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-23T06:08:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25"
    ],
    "keywords": [
      "odd order",
      "semiregular representation",
      "finite groups admitting",
      "regular tournament",
      "regular tmsr"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}