Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Li, Ningyi, Li, Ningyi
Institution: Leiden University
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203708703