{
  "abstract": "Let a finite group $A$ act coprimely on a finite group $G$ and put $C=C_G(A)$. A classical theorem of Burnside asserts that the irreducible characters of a group vanishing nowhere are exactly the linear ones, and Navarro asked in Problem 21.100 of the 21st Kourovka Notebook whether the coprime analogue holds: is the number of $A$-invariant $χ\\in\\Irr(G)$ with $χ_C$ nowhere zero always $|C/C'|$? We answer this negatively. For $A$ cyclic of order $21$ acting on $G=B\\rtimes V$, where $V=\\F_4^2\\oplus\\F_8$ and $B$ is the group of Boolean functions on $V$, the fixed subgroup $C\\cong C_2^{12}$ carries all $4096$ invariant characters while only $1728$ of them are nowhere zero on $C$. Because $C$ is abelian the same example refutes Problem 6.3 of Navarro's problem list, and with it the corresponding statement about the head characters of Isaacs; passing to $G\\rtimes A$ refutes Problem 6.7, a conjecture Isaacs reports is supported by abundant computational evidence. The construction needs only that $V$ have an $A$-stable subset of half its size. This holds for infinitely many pairs $(A,V)$, and for none of dimension below $7$, so the example is minimal over all operator groups of odd order. Exact machine verifications, independent of the proofs, accompany the paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16227",
  "author_total": 1,
  "authors": [
    {
      "name": "Eric Hou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/749rgjbxdca1m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16227",
  "fulltext_url": null,
  "id": "749rgjbxdca1m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16227",
  "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": "W7213405049",
  "pdf_url": "https://arxiv.org/pdf/2609.16227",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/749rgjbxdca1m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16227",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16227"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Coprime Actions and Characters Non-vanishing on the Fixed-point Subgroup",
  "title_text": "Coprime Actions and Characters Non-vanishing on the Fixed-point Subgroup",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/749rgjbxdca1m",
  "venue": "arXiv",
  "versions": []
}