{
  "abstract": "Let $p$ be an odd prime and $G$ a finite $p$-group whose Jennings series has length at most $p$, that is, $D_{p+1}(G)=1$; when $D_p(G)\\ne1$ assume also that $G/D_p(G)$ has nilpotency class at most $p-2$. We prove that $G$ is determined up to isomorphism by its modular group algebra $\\mathbb{F}_p G$, and indeed by the quotient $\\mathbb{F}_p G/\\mathrm{I}^{\\,ν+1}$, where $\\mathrm{I}$ is the augmentation ideal and $ν$ the length of the Jennings series; for an arbitrary finite $p$-group $G$, the quotient $\\mathbb{F}_p G/\\mathrm{I}^{\\,p}$ determines $G/D_p(G)$; and that every automorphism of $\\mathbb{F}_p G$ is the product of an automorphism of $G$ and an automorphism inducing the identity on $\\mathrm{I}/\\mathrm{I}^2$. In particular the modular isomorphism problem has a positive answer for all $p$-groups of exponent $p$ and class less than $p$, and for all $p$-groups of class at most $p-2$ and exponent dividing $p^2$ whose derived subgroup has exponent $p$. Consequently every counterexample in odd characteristic has class at least $p-1$, or a derived subgroup of exponent at least $p^2$, or an element of order $p^3$. The proof combines a descent theorem in the truncated group algebra of a free group, valid for every prime, with a relative rigidity theorem for the unipotent automorphisms of truncated group algebras of groups of exponent $p$, proved through the Lazard correspondence. The bound $p$ is the range in which the logarithm of the group algebra exists.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40868",
  "author_total": 1,
  "authors": [
    {
      "name": "Diego Tuzzolo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cgmmqkzacjqzc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40868",
  "fulltext_url": null,
  "id": "cgmmqkzacjqzc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40868",
  "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/2610.40868",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cgmmqkzacjqzc.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40868",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40868"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Short Jennings series and the modular isomorphism problem",
  "title_text": "Short Jennings series and the modular isomorphism problem",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/cgmmqkzacjqzc",
  "venue": "arXiv",
  "versions": []
}