{
  "abstract": "Let $P$ be a $p$-group that acts faithfully and coprimely on a finite group $H$. A classical result of D.\\ S.\\ Passman, in the form proved by I.\\ M.\\ Isaacs, asserts that there exists an element $x\\in H$ such that $|C_P(x)|\\le (|P|/p)^{1/p}$. In this paper, we study the corresponding question for the action of $P$ on the conjugacy classes of $H$. We show that for every odd prime $p$ the analogue of Isaacs' bound fails for classes. On the other hand, we prove that if $H$ is solvable, then there always exists a conjugacy class $c$ of $H$ whose stabilizer in $P$ has order at most $|P|^{1/2}$, and that this bound is best possible for every prime $p$. As a consequence, there always exists an irreducible character of $H$ whose inertia group in $P$ has order at most $|P|^{1/2}$. The proof proceeds via a result on linear actions of $p$-nilpotent groups, and in the quasiprimitive case it relies on a classification of Holt and Yang and on computer calculations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05904",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Michael Keller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ffyb218s1s3c3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05904",
  "fulltext_url": null,
  "id": "ffyb218s1s3c3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05904",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ffyb218s1s3c3.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Large orbits of $p$-groups on conjugacy classes",
  "title_text": "Large orbits of p-groups on conjugacy classes",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:08Z",
  "url": "https://research.draconitic.com/paper/ffyb218s1s3c3",
  "venue": "arXiv",
  "versions": []
}