On Zn⋊Z2-Hopf-Galois structures

Abstract

Let be a finite Galois extension of fields with . In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. The dihedral group is one particular example of a semidirect product of and . In this article we count the number of Hopf-Galois structures with Galois group Γ of type G, where are groups of the form when n is odd with radical of n being a Burnside number. As an application we also find the corresponding number of skew braces.