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

@misc{li2026exponents,
  title = {{Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras}},
  author = {Ningyi Li and Ningyi Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.14972},
  eprint = {2608.14972},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14972},
}

TY  - UNPB
TI  - Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras
AU  - Ningyi Li
AU  - Ningyi Li
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2608.14972
UR  - https://arxiv.org/abs/2608.14972
AB  - Let G=FΓ be a factorization of a finite group, with neither factor assumed normal and with F∩Γ allowed to be nontrivial. We prove that exp(G) divides 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 Rep(H⊗_kk̅) 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.
ER  -