发表机构
Leiden University（莱顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对有限群的因子分解问题，证明了相关指数整除性，进而解决了 Kashina 指数猜想，还推导了低维 Hopf 代数等领域的应用。
AI 中文摘要
设有限群 $G=F\Gamma$ 为因子分解，其中任一因子均非正规，且允许 $F\cap\Gamma$ 非平凡。我们证明 $\exp(G)$ 整除 $\operatorname{lcm}(|F|,|\Gamma|)$，等价于 $\gcd([G:F],[G:\Gamma])\exp(G)$ 整除 $|G|$，这回答了 Natale 提出的上同调整除问题。结合群论整除性、Natale 的指数界与提升论证，我们对域 $k$ 上满足 $\operatorname{Rep}(H\otimes_k\overline{k})$ 为群论的任意有限维半单余半单 Hopf 代数 $H$，证明了 Etingof 与 Gelaki 提出的任意域形式下的 Kashina 指数猜想。同一论证还证明了以任意 $G$-模为系数的对应三阶上同调整除性。对复群论范畴，在未假设纤维函子的情况下，我们还建立了上同调分解假设下的 Frobenius-Schur 指数整除性，并推导了其在低维 Hopf 代数与阿贝尔扩张中的应用。
英文摘要
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.
CommentsThis paper has been withdrawn at the request of one of the co-authors
