{
  "abstract": "Let $p$ be an odd prime and $H$ a finite $p$-group. Hertweck and Soriano parametrize the central Frattini extensions of $H$ by subspaces of a relation module $V$, matching isomorphism classes of extensions with orbits of $\\mathrm{Out}(H)$ and isomorphism classes of the associated small group rings with orbits of $\\mathrm{Out}(\\mathbb{F}_p H)$. The discrepancy between the two is measured by the group $U(H)\\leq GL(V)$ of displacements induced by the normalized unipotent automorphisms of $\\mathbb{F}_p H$; it is the mechanism behind the counterexamples to the modular isomorphism problem in characteristic $2$. We prove that for $H$ of exponent $p$ and class at most $p-2$ every such automorphism acts on $V$ exactly as an automorphism of $H$ does, and that $U(H)=\\expΩ\\bigl(\\mathrm{Der}_{>0}(\\mathfrak{l}(H))\\bigr)$, where $\\mathfrak{l}(H)$ is the Lazard Lie algebra of $H$, $\\mathrm{Der}_{>0}$ its derivations raising the lower central filtration, and $Ω$ the natural action of derivations on $H_2(\\mathfrak{l}(H))$. Hence $U(H)=1$ precisely when every derivation of positive degree acts trivially on $H_2(\\mathfrak{l}(H))$. We compute $U(H)$ for all groups of class $2$ and exponent $p$, obtaining $p^{2n(n-1)}$ for the Heisenberg groups $\\mathrm{Heis}(p^n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06909",
  "author_total": 1,
  "authors": [
    {
      "name": "Diego Tuzzolo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1mrhw2vhwpd1c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06909",
  "fulltext_url": null,
  "id": "1mrhw2vhwpd1c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06909",
  "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": "W7221043527",
  "pdf_url": "https://arxiv.org/pdf/2610.06909",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1mrhw2vhwpd1c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.06909",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06909"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra",
  "title_text": "Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/1mrhw2vhwpd1c",
  "venue": "arXiv",
  "versions": []
}