{
  "abstract": "Let $G=FΓ$ be a factorization of a finite group, with neither factor assumed normal and with $F\\capΓ$ allowed to be nontrivial. We prove that $\\exp(G)$ divides $\\operatorname{lcm}(|F|,|Γ|)$, or equivalently that $\\gcd([G:F],[G:Γ])\\exp(G)$ divides $|G|$. This answers a cohomological divisibility question posed by Natale. Combining the group-theoretic divisibility with Natale's exponent bound and a lifting argument, we prove Kashina's exponent conjecture, in the arbitrary-field formulation of Etingof and Gelaki, for every finite-dimensional semisimple and cosemisimple Hopf algebra $H$ over a field $k$ for which $\\operatorname{Rep}(H\\otimes_k\\overline{k})$ is group-theoretical. The same argument proves the corresponding degree-three cohomological divisibility for coefficients in an arbitrary $G$-module. For complex group-theoretical categories, we also establish Frobenius-Schur exponent divisibility under a cohomological factorization hypothesis, without assuming a fiber functor. We derive applications to low-dimensional Hopf algebras and abelian extensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14972",
  "author_total": 2,
  "authors": [
    {
      "name": "Ningyi Li"
    },
    {
      "name": "Ningyi Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f0rcznhr8mn2j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14972",
  "fulltext_url": null,
  "id": "f0rcznhr8mn2j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14972",
  "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"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7203708703",
  "pdf_url": "https://arxiv.org/pdf/2608.14972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f0rcznhr8mn2j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2608.14972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras",
  "title_text": "Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/f0rcznhr8mn2j",
  "venue": "arXiv",
  "versions": []
}