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.