{
  "abstract": "For every odd prime $p$, we construct a finite left brace $A_p$ of order $p^{2p+1}$ whose additive group is elementary abelian and whose multiplicative group $G_p$ is a powerful $p$-group, but such that $A_p$ is not right nilpotent. More precisely, $G_p'=G_p^p\\cong C_p^2, \\operatorname{cl}(G_p)=2, \\exp(G_p)=p^2,$ and $\\operatorname{Soc}(A_p)=0$. Thus powerfulness does not force right nilpotence even for multiplicative groups of class two with derived subgroup of order $p^2$. The obstruction is explicit: $A_p$ contains a three-dimensional trivial ideal $T$ satisfying $T*A_p=T$, whereas its left series is the ideal-power filtration of a nilpotent commutative algebra. Both $T$ and $A_p/T$ are right nilpotent, so right nilpotence of finite left braces is not closed under extensions. The construction is uniform and arises from a finite local commutative algebra, a square-zero derivation, and an invariant group homomorphism, yielding a regular affine subgroup as the kernel of a homomorphism on a semidirect product. It disproves the Shalev--Smoktunowicz conjecture in every odd characteristic and yields finite non-degenerate irretractable involutive set-theoretic solutions of the Yang--Baxter equation whose permutation groups are powerful $p$-groups of class two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.01884",
  "author_total": 1,
  "authors": [
    {
      "name": "Brecht F. Verbeken"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c50w7gygfn5xp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.01884",
  "fulltext_url": null,
  "id": "c50w7gygfn5xp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.01884",
  "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",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7172439935",
  "pdf_url": "https://arxiv.org/pdf/2608.01884",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c50w7gygfn5xp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2608.01884",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.01884"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Powerful multiplicative groups do not force right nilpotence in finite braces",
  "title_text": "Powerful multiplicative groups do not force right nilpotence in finite braces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/c50w7gygfn5xp",
  "venue": "arXiv",
  "versions": []
}