Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group

TL;DR

This paper characterizes all groups N for which a cyclic group G admits a Hopf-Galois structure, extending previous work that fixed N as cyclic and now focusing on fixed G.

Contribution

It provides a complete characterization of groups N that admit a Hopf-Galois structure with fixed cyclic G, complementing prior results that fixed N as cyclic.

Findings

Characterization of groups N for fixed cyclic G

Extension of Rump's previous results

Complete classification of realizable pairs (G,N)

Abstract

Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete characterization of the groups $N$ for which $(G,N)$ is realizable.