{
  "abstract": "A group is called an $n$-$\\bfc$-group if each of its conjugacy classes is finite and contains at most $n$ elements. By a theorem of B.~H.~Neumann the derived subgroup of such a group is finite of $n$-bounded order, whereas the order of the group itself need not be bounded, as extraspecial $p$-groups show. We prove that an $n$-$\\bfc$-group with trivial centre is finite of order at most $n^{64(\\log n)^{5}}$. Passing to the quotient by the hypercentre, we deduce that no assumption on the structure of the group is needed at all: for an arbitrary finite $n$-$\\bfc$-group $G$ one has $|G:Z_\\infty(G)|<n^{64(\\log n)^{5}}$. The finiteness of the group is required here only for the correct definition of the hypercentre.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31387",
  "author_total": 1,
  "authors": [
    {
      "name": "Ilya Gorshkov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a4bxq3vkwdahy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31387",
  "fulltext_url": null,
  "id": "a4bxq3vkwdahy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31387",
  "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": "W7214611033",
  "pdf_url": "https://arxiv.org/pdf/2609.31387",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a4bxq3vkwdahy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31387",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31387"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the order of a finite group with trivial centre and bounded conjugacy classes",
  "title_text": "On the order of a finite group with trivial centre and bounded conjugacy classes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/a4bxq3vkwdahy",
  "venue": "arXiv",
  "versions": []
}