Nonabelian Case of Hopf Galois Structures on Nonnormal Extensions of Degree pqw

Abstract

We look at Hopf Galois structures with square free pqw degree on separable field extensions (nonnormal) L/K. Where E/K is the normal closure of L/K, the group permutation of degree pqw is G = Gal(E/K). We study details of the nonabelian case, where Jl = ⟨σ, [τ, αl ]⟩ is a nonabelian regular subgroup of Hol(N) for 1 ≤ l ≤ w − 1. We first find the group permutation G, and then the Hopf Galois structures for each G. In this case, there exists four G such that the Hopf Galois structures are admissible within the field extensions L/K.