Solubility Criteria for Hopf-Galois Structures

Let $L/K$ be a finite Galois extension of fields with group $\Gamma$. Associated to each Hopf-Galois structure on $L/K$ is a group $G$ of the same order as the Galois group $\Gamma$. The type of the Hopf-Galois structure is by definition the isomorphism type of $G$. We investigate the extent to which general properties of either of the groups $\Gamma$ and $G$ constrain those of the other. Specifically, we show that if $G$ is nilpotent then $\Gamma$ is soluble, and that if $\Gamma$ is abelian then $G$ is soluble. The proof of the latter result depends on the classification of finite simple groups. In contrast to these results, we give some examples where the groups $\Gamma$ and $G$ have different composition factors. In particular, we show that a soluble extension may admit a Hopf-Galois structure of insoluble type.