Hopf-Galois Realizability of $\mathbb{Z}_n\rtimes\mathbb{Z}_2$

TL;DR

This paper characterizes when groups of the form rac{Z}_n times rac{Z}_2 can be realized as Hopf-Galois structures, providing necessary and sufficient conditions related to Burnside numbers.

Contribution

It establishes necessary and sufficient conditions for Hopf-Galois realizability of groups rac{Z}_n times rac{Z}_2, classifying associated skew braces.

Findings

Necessary conditions for realizability are derived.

Sufficient conditions are proven when n's radical is a Burnside number.

Complete classification of certain skew braces is achieved.

Abstract

Let $G$ and $N$ be finite groups of order $2n$ where $n$ is odd. We say the pair $(G,N)$ is Hopf-Galois realizable if $G$ is a regular subgroup of $\h(N)=N\rtimes\au(N)$. In this article we give necessary conditions on $G$ (similarly $N$) when $N$ (similarly $G$) is a group of the form $\mathbb{Z}_n\rtimes\mathbb{Z}_2$. Further we show that this condition is also sufficient if radical of $n$ is a Burnside number. This classifies all the skew braces which has the additive group (or the multiplicative group) to be isomorphic to $\mathbb{Z}_n\rtimes\mathbb{Z}_2$, in this case.