Skew left braces and isomorphism problems for Hopf–Galois structures on Galois extensions

Abstract

Given a finite group [Formula: see text], we study certain regular subgroups of the group of permutations of [Formula: see text], which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to [Formula: see text] and Hopf–Galois structures admitted by a Galois extension of fields with Galois group isomorphic to [Formula: see text]. We study the questions of when two such subgroups yield isomorphic skew left braces or Hopf–Galois structures involving isomorphic Hopf algebras. In particular, we show that in some cases the isomorphism class of the Hopf algebra giving a Hopf–Galois structure is determined by the corresponding skew left brace. We investigate these questions in the context of a variety of existing constructions in the literature. As an application of our results we classify the isomorphically distinct Hopf algebras that give Hopf–Galois structures on a Galois extension of degree [Formula: see text] for [Formula: see text] prime numbers.