On [formula omitted]-Hopf-Galois structures

Abstract

Let K/F be a finite Galois extension of fields with Gal(K/F)=Γ. 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 Zn and Z2. In this article we count the number of Hopf-Galois structures with Galois group Γ of type G, where Γ,G are groups of the form Zn⋊ϕZ2 when n is odd with radical of n being a Burnside number. As an application we also find the corresponding number of skew braces.