{
  "abstract": "Let $G$ be a finite group and let $p$ be a prime. We prove that if the common-divisor graph on the nontrivial conjugacy-class sizes of $p$-regular elements of $G$ is disconnected, then $G$ is soluble. This settles the remaining case identified by Camina and collaborators. Their structure results provide an abelian normal $p'$-subgroup $M$. We choose a noncentral $r$-element $x\\in M$, with $r\\ne p$, such that $[G:C_G(x)]$ is coprime to $pr$, and show that $C_G(x)/M$ is a $\\{p,r\\}$-group. Thus $C_G(x)/M$ is a Hall $\\{p,r\\}$-subgroup of $G/M$. The separation between the two graph components then implies that every $p$-regular conjugacy-class size in $G/M$ has all its prime divisors in $\\{p,r\\}$. A theorem of Dolfi and Lucido, together with Burnside's $p^a q^b$-theorem, yields the solubility of $G/M$ and hence of $G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27960",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhi-Lin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fjqfdnxhtrmkf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27960",
  "fulltext_url": null,
  "id": "fjqfdnxhtrmkf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27960",
  "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.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214160249",
  "pdf_url": "https://arxiv.org/pdf/2609.27960",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fjqfdnxhtrmkf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2609.27960",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27960"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Solubility from a disconnected common-divisor graph on $p$-regular conjugacy-class sizes",
  "title_text": "Solubility from a disconnected common-divisor graph on p-regular conjugacy-class sizes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/fjqfdnxhtrmkf",
  "venue": "arXiv",
  "versions": []
}