{
  "abstract": "Let $G$ be a finite group, let $ρ(G)$ denote the number of distinct prime divisors of $|G|$, and let $σ(G)$ be the maximal number of distinct prime divisors of the order of an element of $G$. Let $ρ(n)=\\max\\{ρ(G) \\mid G \\text{ solvable}, σ(G)=n\\}$. We prove that $ρ(n)\\le 4n-1+π(2n)$ for all $n$, and, with the help of a computer, that $ρ(n)\\le 9n/2$ for all $n\\le 16$. In particular, $ρ(n)\\le\\lfloor 9n/2\\rfloor+2$ and $ρ(n)\\le (101/22)n$ for all $n$, and $ρ(n)\\le 9n/2$ for all $n\\ge 58$. This improves the bound $ρ(n)\\le (17/3)n$ due to Yang, and it shows that the bound $ρ(n)\\le 5n$ stated by Bellotti, Keller and Trudgian holds, although its original proof contains a gap. Moreover, we show that $ρ(n)\\ge 3n$ for all $n\\ge 4$, which proves one half of a conjecture of the second author stating that $ρ(n)=3n$ for $n\\ge 4$. We also derive a consequence for character codegrees and, conditionally, for arbitrary finite groups. The proofs of the main results do not rely on the classification of finite simple groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25571",
  "author_total": 2,
  "authors": [
    {
      "name": "Enrico Jabara"
    },
    {
      "name": "Thomas Michael Keller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/27598hqwdrfyd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25571",
  "fulltext_url": null,
  "id": "27598hqwdrfyd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25571",
  "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": "W7223071533",
  "pdf_url": "https://arxiv.org/pdf/2610.25571",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/27598hqwdrfyd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25571",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25571"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Huppert's $ρ$-$σ$ conjecture for primes in the orders of elements",
  "title_text": "On Huppert's ρ-σ conjecture for primes in the orders of elements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/27598hqwdrfyd",
  "venue": "arXiv",
  "versions": []
}