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

TL;DR

This paper classifies finite groups for which the Hopf--Galois correspondence is bijective across all structures, extending previous work on Galois extensions and skew braces to a broader group classification.

Contribution

It provides a classification of groups N where the Hopf--Galois correspondence is bijective for all structures of type N on Galois extensions.

Findings

Classified groups N with bijective Hopf--Galois correspondence for all structures

Extended previous classifications from Galois groups to arbitrary groups N

Connected the properties of groups N with the bijectivity of the correspondence

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.