Classification of the types for which every Hopf--Galois correspondence is bijective

Abstract

Let $L/K$ be any finite Galois extension with Galois group $G$. It is known by Chase and Sweedler that the Hopf--Galois correspondence is injective for every Hopf--Galois structure on $L/K$, but it need not be bijective in general. Hopf--Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of $L/K$ in the image of the Hopf--Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups $G$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure on any $G$-Galois extension. In this paper, using a similar approach, we shall classify the groups $N$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure of type $N$ on any Galois extension.

10 pages; modified the definitions of H in Examples 3.1 to 3.5 based on referee's suggestions

References in corpus (4)

• From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures
• On bi-skew braces and brace blocks
• On the connection between Hopf--Galois structures and skew braces
• Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group