{
  "abstract": "Let $G$ be a finite group and let $V(\\mathbb ZG)$ be the group of normalized units of its integral group ring. We prove that every $N$-prime arc of $V(\\mathbb ZG)$ either already occurs in $G$ or admits commuting witnesses of distinct prime orders. Writing $A(Δ)$ for the arc set of a directed graph $Δ$, $E(Δ)$ for the edge set of an undirected graph, and $\\operatorname{Sym}(E)$ for the two orientations of the edges in $E$, this is equivalent to the exact formula \\[ A\\bigl(Γ_{\\mathrm N}(V(\\mathbb ZG))\\bigr) = A\\bigl(Γ_{\\mathrm N}(G)\\bigr) \\cup \\operatorname{Sym}\\!\\bigl( E(Γ_{\\mathrm{GK}}(V(\\mathbb ZG))) \\bigr). \\] Consequently, the $N$-Prime Graph Question has an affirmative answer for $G$ if and only if the Prime Graph Question does.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.29105",
  "author_total": 1,
  "authors": [
    {
      "name": "Brecht F. Verbeken"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ds3bfj9sedx1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.29105",
  "fulltext_url": null,
  "id": "8ds3bfj9sedx1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.29105",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7172353409",
  "pdf_url": "https://arxiv.org/pdf/2607.29105",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ds3bfj9sedx1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2607.29105",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.29105"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The N-Prime Graph Question is equivalent to the Prime Graph Question",
  "title_text": "The N-Prime Graph Question is equivalent to the Prime Graph Question",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/8ds3bfj9sedx1",
  "venue": "arXiv",
  "versions": []
}