Let be a group of order where is prime and . We give a strategy to enumerate the regular subgroups of normalized by the left representation of . These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions with . We prove that every such regular subgroup is contained in the normalizer in of the -Sylow subgroup of . 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 , where is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.
Timothy Kohl. "Regular permutation groups of order mp and Hopf Galois structures." Algebra Number Theory 7 (9) 2203 - 2240, 2013. https://doi.org/10.2140/ant.2013.7.2203