Groups of order 4 p, twisted wreath products and Hopf–Galois theory

Abstract

The work of Greither and Pareigis details the enumeration of the Hopf–Galois structures (if any) on a given separable field extension. We consider the cases where L / K is already classically Galois with Γ = Gal ( L / K ) , where | Γ | = 4 p for p > 3 a prime. The goal is to determine those regular (transitive and fixed point free) subgroups N of Perm ( Γ ) that are normalized by the left regular representation of Γ. A key fact that aids in this search is the observation that any such regular subgroup, necessarily of order 4 p, has a unique subgroup of order p. This allows us to show that all such N are contained in a ‘twisted’ wreath product, a subgroup of high index in Perm ( Γ ) which has a very computationally convenient description that allows us to perform the aforementioned enumeration.