Regular permutation groups of order mp and Hopf Galois structures

Abstract

Let Γ be a group of order mp where p is prime and p>m. We give a strategy to enumerate the regular subgroups of Perm(Γ) normalized by the left representation λ(Γ) of Γ. These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions L∕K with Γ= Gal(L∕K). We prove that every such regular subgroup is contained in the normalizer in Perm(Γ) of the p-Sylow subgroup of λ(Γ). 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(p−1), where p is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.