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

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.

Fixed some typos

References in corpus (2)

• Bi-skew braces and Hopf Galois structures
• Solubility Criteria for Hopf-Galois Structures

Cited by in corpus (3)

• On the connection between Hopf--Galois structures and skew braces
• Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension
• Classification of the types for which every Hopf--Galois correspondence is bijective