{
  "abstract": "Given a finite transitive group $G\\leq \\operatorname{Sym}(Ω)$, the derangement graph $Γ_G$ is the graph whose vertex set is $G$, and two vertices $g$ and $h$ are adjacent if the ratio $h^{-1}g$ is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement graph of the transitive permutation group corresponding to the natural action of $\\operatorname{PGL}_{2}(q)$ on the projective line $\\operatorname{PG}_{1}(q)$ is \\begin{align*} \\operatorname{Aut}(Γ_{\\operatorname{PGL}_{2}({q})}) = \\left(L_{\\operatorname{PGL}_2(q)}\\times R_{\\operatorname{PGL}_2(q)}\\right) \\rtimes \\left(\\langle ψ\\rangle \\times γ_{\\operatorname{Aut}(\\mathbb{F}_q)}\\right), \\end{align*} where $L_{\\operatorname{PGL}_2(q)}$ is the left-regular representation of $\\operatorname{PGL}_2(q)$, $R_{\\operatorname{PGL}_2(q)}$ is the right-regular representation of $\\operatorname{PGL}_2(q)$, $γ_{\\operatorname{Aut}(\\mathbb{F}_q)}$ is the group of conjugation by elements of $\\operatorname{Aut}(\\mathbb{F}_q)$, and $ψ: \\operatorname{PGL}_2(q) \\to \\operatorname{PGL}_2(q)$ such that $ψ(x) = x^{-1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32969",
  "author_total": 1,
  "authors": [
    {
      "name": "Andriaherimanana Sarobidy Razafimahatratra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4hxrqxpm7q9be.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32969",
  "fulltext_url": null,
  "id": "4hxrqxpm7q9be",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32969",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214772169",
  "pdf_url": "https://arxiv.org/pdf/2609.32969",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4hxrqxpm7q9be.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32969",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32969"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The automorphism group of the derangement graph of $\\operatorname{PGL}_{2}(q)$ acting on the projective line",
  "title_text": "The automorphism group of the derangement graph of PGL_2(q) acting on the projective line",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4hxrqxpm7q9be",
  "venue": "arXiv",
  "versions": []
}