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

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 $\mathrm{lcm}(|F|,|\Gamma |)$, 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 $\mathrm{Rep}(H{\otimes }_k\overline{k})$ 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.

12 pages