{
  "abstract": "For a finite group $G$, let $ω(G)$ be the largest size of a pairwise noncommuting set and put $z(G)=[G:\\mathrm Z(G)]$. We determine the maximum possible central index under $ω(G)\\le k$ for every integer $k\\ge1$, giving an exact answer to Neumann's central-index question and sharpening Pyber's exponential upper bound. The maximum is $1$ for $k=1,2$, $(k-1)^2$ for $3\\le k\\le6$, $81$ for $k=7,8$, and $4^{\\lfloor(k-1)/2\\rfloor}$ for $k\\ge9$. In the last range, with $r=\\lfloor(k-1)/2\\rfloor$, equality holds exactly when $G'\\cong C_2$ and $G/\\mathrm Z(G)\\cong\\mathbb F_2^{2r}$. We also obtain the sharp universal bound $z(G)\\le(81/64)2^{ω(G)-1}$. All maximizers are classified up to isoclinism. The results extend to arbitrary groups with finite clique number through finite isoclinic representatives.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00943",
  "author_total": 1,
  "authors": [
    {
      "name": "Guillaume Lecomte"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6136tbnc698pj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00943",
  "fulltext_url": null,
  "id": "6136tbnc698pj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00943",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00943",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6136tbnc698pj.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00943",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00943"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Exact Central-Index Profile of Groups with Bounded Noncommutativity",
  "title_text": "The Exact Central-Index Profile of Groups with Bounded Noncommutativity",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/6136tbnc698pj",
  "venue": "arXiv",
  "versions": []
}