Fixed-point free pairs of homomorphisms and nonabelian Hopf-Galois structure

Given finite groups Γ and G of order n, regular embeddings from Γ to the holomorph of G yield Hopf-Galois structures on a Galois extension L|K of fields with Galois group Γ. Here we consider regular embeddings that arise from fixed-point free pairs of homomorphisms from Γ to G. If G is a complete group, then all regular embeddings arise from fixed-point free pairs. For all Γ, G of order n = p(p-1) with p a safeprime, we compute the number of Hopf-Galois structures that arise from fixed-point free pairs, and compare the results with a count of all Hopf-Galois structures obtained by T. Kohl. Using the idea of fixed-point free pairs, we characterize the abelian Galois groups Γ of even order or order a power of p, an odd prime, for which L|K admits a nonabelian Hopf Galois structure. The paper concludes with some new classes of abelian groups Γ for which every Hopf-Galois structure has type Γ (and hence is abelian).