{
  "abstract": "For a subset $S$ of a finite group $G$, let $I_G(S)$ denote the set of commutators $[g,x]=g^{-1}g^x$, where $g\\in G$ and $x\\in S$. Suppose that a finite group $G$ has a Carter subgroup $C$, that is, a nilpotent subgroup containing its normalizer. Suppose that any subgroup generated by a subset of $I_G(C)$ is $r$-generated. We prove that if $G$ is soluble, then the derived subgroup $G'$ has $r$-bounded rank. We produce examples showing that the solubility condition cannot be dropped. For any finite group, we prove that the rank of $G'$ is $(r,l)$-bounded, where $l$ is the maximum rank of composition factors of $G$ isomorphic to $PSL_2(q)$ for $q\\equiv 7\\,(\\operatorname{mod}8)$. We also prove in the general case that $G'$ has $r$-bounded rank under the additional condition that for any $x\\in I_G(C)$, any subgroup generated by a subset of $I_G(x)$ is $r$-generated. The proofs rely on the classification of finite simple groups, using which we prove that if a finite simple group $G$ has an element $x$ of prime order $p$ such that any subgroup generated by a subset of $I_G(x)$ is $r$-generated, then $G$ has $r$-bounded (Prüfer) rank or is isomorphic to $PSL_2(q)$ with $q \\equiv 3\\,(\\operatorname{mod}4)$ when $p=2$, or to $PSL_2(q)$ with $q \\equiv -1\\,(\\operatorname{mod} p)$ when $p\\ne 2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28309",
  "author_total": 4,
  "authors": [
    {
      "name": "Cristina Acciarri"
    },
    {
      "name": "Robert M. Guralnick"
    },
    {
      "name": "Evgeny Khukhro"
    },
    {
      "name": "Pavel Shumyatsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8ajz0ndftzs1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28309",
  "fulltext_url": null,
  "id": "e8ajz0ndftzs1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28309",
  "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",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214151704",
  "pdf_url": "https://arxiv.org/pdf/2609.28309",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8ajz0ndftzs1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28309",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28309"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Rank properties of commutators in finite groups",
  "title_text": "Rank properties of commutators in finite groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/e8ajz0ndftzs1",
  "venue": "arXiv",
  "versions": []
}