{
  "abstract": "Let $G$ be a finite nonabelian group of odd order, let $p$ be the smallest prime divisor of $|G|$, and let $V$ be a finite faithful completely reducible $G$-module, possibly of mixed characteristic. Suppose that $M$ is the largest orbit size in the action of $G$ on $V$. It is known that $|G:G'|\\le M$, and that equality holds only for abelian groups if $|G|$ is odd. We prove that $|G:G'|\\le M/p$, unless $p=3$ and $G$ is the direct product of an abelian group and a group isomorphic to $Γ(2^3)$ or to $Γ(2^3)\\times C_3$ acting on $V$ in a specific way, in which case $|G:G'|=3M/7$. In particular, $|G:G'|\\le 3M/7$ for every nonabelian group of odd order. If $|V|$ is odd, then $|G:G'|=M/p$ holds if and only if $|G'|=p$; more precisely, nilpotent groups then have regular orbits, whereas non-nilpotent groups satisfy $|G:G'|\\le 3M/13$, and even $|G:G'|<M/(2p)$ if $p\\ge5$. The bound $3M/13$ is best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08690",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Michael Keller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2zzyyqdtfe7cy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08690",
  "fulltext_url": null,
  "id": "2zzyyqdtfe7cy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08690",
  "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": "W7221011226",
  "pdf_url": "https://arxiv.org/pdf/2610.08690",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2zzyyqdtfe7cy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08690",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08690"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Abelian quotients and the largest orbit sizes of linear groups of odd order",
  "title_text": "Abelian quotients and the largest orbit sizes of linear groups of odd order",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/2zzyyqdtfe7cy",
  "venue": "arXiv",
  "versions": []
}