On $\mathbb{Z}_N\rtimes\mathbb{Z}_2$-Hopf-Galois structures

TL;DR

This paper counts Hopf-Galois structures and skew braces for Galois extensions with groups of the form rac{rac{N}{ times times ext{Burnside number}}.

Contribution

It extends previous work by enumerating Hopf-Galois structures for groups of the form rac{rac{N}{ times times ext{Burnside number}} and finds related skew braces.

Findings

Counted the number of Hopf-Galois structures for specified groups.

Established a link between Hopf-Galois structures and skew braces.

Extended enumeration to groups with radical of N being a Burnside number.

Abstract

Let $K/F$ be a finite Galois extension of fields with $Gal(K/F)=\Gamma$. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. Dihedral group is one particular example of semidirect product of $\mathbb{Z}_n$ and $\mathbb{Z}_2$. In this article we count the number of Hopf-Galois structures with Galois group $\Gamma$ of type $G$, where $\Gamma,G$ are groups of the form $\mathbb{Z}_N\rtimes_\phi\mathbb{Z}_2$ when $N$ is odd with radical of $N$ being a Burnside number. As an application we also find the corresponding number of skew braces.