Regular permutation groups of order mp and Hopf Galois structures

Abstract

Let $\Gamma$ be a group of order $mp$ where $p$ is prime and $p>m$. We give a strategy to enumerate the regular subgroups of $Perm\left(\Gamma \right)$ normalized by the left representation $\lambda \left(\Gamma \right)$ of $\Gamma$. These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions $L∕K$ with $\Gamma =Gal\left(L∕K\right)$. We prove that every such regular subgroup is contained in the normalizer in $Perm\left(\Gamma \right)$ of the $p$-Sylow subgroup of $\lambda \left(\Gamma \right)$. This normalizer has an affine representation that makes feasible the explicit determination of regular subgroups in many cases. We illustrate our approach with a number of examples, including the cases of groups whose order is the product of two distinct primes and groups of order $p\left(p-1\right)$, where $p$ is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.