Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras

[2608.14972] Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Quantum Algebra arXiv:2608.14972 (math) [Submitted on 15 Aug 2026] Title: Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras Authors: Ningyi Li (Leiden University), Ningyi Li (University of Padova) View a PDF of the paper titled Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras, by Ningyi Li (Leiden University) and Ningyi Li (University of Padova) View PDF HTML (experimental) Abstract: Let $G=F\Gamma$ be a factorization of a finite group, with neither factor assumed normal and with $F\cap\Gamma$ allowed to be nontrivial. We prove that $\exp(G)$ divides $\operatorname{lcm}(|F|,|\Gamma|)$, or equivalently that $\gcd([G:F],[G:\Gamma])\exp(G)$ 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. Comments: 12 pages Subjects: Quantum Algebra (math.QA) ; Group Theory (math.GR) MSC classes: 16T05, 20D40 (Primary), 18M20, 20J06 (Secondary) Cite as: arXiv:2608.14972 [math.QA] (or arXiv:2608.14972v1 [math.QA] for this version) https://doi.org/10.48550/arXiv.2608.14972 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ningyi Li [ view email ] [v1] Sat, 15 Aug 2026 01:47:23 UTC (14 KB) Full-text links: Access Paper: View a PDF of the paper titled Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras, by Ningyi Li (Leiden University) and Ningyi Li (University of Padova) View PDF HTML (experimental) TeX Source view license Current browse context: math.QA < prev | next > new | recent | 2026-08 Change to browse by: math math.GR References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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