{
  "abstract": "For a finite group $G$, let $\\mathrm W(G)$ be the intersection of the normalizers of the nilpotent residuals of the subnormal subgroups of $G$, and let $\\ell_{\\mathrm W}(G)$ be the length of the associated upper series. Thus $\\mathrm W(G)=G$ exactly when all these residuals are normal in $G$. We prove that every finite soluble group satisfies $\\ell_{\\mathrm W}(G)\\le h(G)\\le4\\ell_{\\mathrm W}(G)$, where $h(G)$ is the Fitting height, and that $h(G)\\le3\\ell_{\\mathrm W}(G)$ if $|G|$ is odd or the Sylow $2$-subgroups of $G$ are abelian. Both constants are attained at every length, the constant $4$ by $\\{2,3\\}$-groups. Moreover, every pair $(n,h)$ with $n\\le h\\le4n$ occurs as $(\\ell_{\\mathrm W}(G),h(G))$. The constant $4$ comes from primitive groups. A soluble primitive group $V\\rtimes M$ satisfies $\\mathrm W(V\\rtimes M)=V\\rtimes M$ if and only if every abelian subgroup of $\\mathrm F(M)$ is cyclic. It then has Fitting height four if and only if $\\mathrm F(M)=C\\times Q_8$, with $C$ cyclic, and $M$ induces the full group $\\mathrm{Out}(Q_8)\\cong S_3$. We also determine when the base group of a soluble regular wreath product is the first term of the series.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28082",
  "author_total": 3,
  "authors": [
    {
      "name": "Lü Gong"
    },
    {
      "name": "Yan Yan"
    },
    {
      "name": "Enric Cosme Llópez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0f88177bmwwdg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28082",
  "fulltext_url": null,
  "id": "0f88177bmwwdg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28082",
  "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": "W7223125955",
  "pdf_url": "https://arxiv.org/pdf/2610.28082",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0f88177bmwwdg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28082",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28082"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Wielandt subgroups of nilpotent residuals and Fitting-height bounds",
  "title_text": "Wielandt subgroups of nilpotent residuals and Fitting-height bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/0f88177bmwwdg",
  "venue": "arXiv",
  "versions": []
}