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

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