{
  "abstract": "Many numerical invariants of a finite group arise as the multiplicity of the trivial character in a naturally associated class function, and the remaining multiplicities carry finer structural information. For the number of conjugacy classes of elements whose order involves only a chosen set of primes, the natural such class function is a generalized character interpolating between the classical conjugating character and the regular character. We prove that it is a character in two important general cases, obtained independently by Robinson, and we rule out a family of groups and characters in which, we argue, a counterexample would be most likely to arise. Furthermore, we characterize when these class functions are compatible with passing to a subgroup and thereby sharpen a theorem of Sangroniz. The analogous class function for real elements is shown to always give a character. Finally, we give a dual construction that, together with the recently proved McKay Conjecture, yields a best-possible criterion for a finite group to have an abelian Sylow $p$-subgroup.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26464",
  "author_total": 1,
  "authors": [
    {
      "name": "Christopher A. Schroeder"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpjt50kdkqjbf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26464",
  "fulltext_url": null,
  "id": "fpjt50kdkqjbf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26464",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214100705",
  "pdf_url": "https://arxiv.org/pdf/2609.26464",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpjt50kdkqjbf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26464",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26464"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Refining invariants of finite groups with class functions",
  "title_text": "Refining invariants of finite groups with class functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/fpjt50kdkqjbf",
  "venue": "arXiv",
  "versions": []
}