Constructing Hopf-Galois structures and skew bracoids of small degree

TL;DR

This paper develops algorithms to classify and enumerate Hopf-Galois structures and skew bracoids of small degree, extending existing results and exploring structures of degree 2pq with p,q odd primes.

Contribution

It introduces algorithms based on the connection to transitive subgroups of the holomorph, enabling classification of these structures for small degrees and degree 2pq.

Findings

Algorithms classify and enumerate structures for small degrees.

Extended classifications to degree 2pq where p,q are odd primes.

Made enumeration-based observations and proposed a conjecture.

Abstract

Using the fact that Hopf-Galois structures on separable extensions and skew bracoids are both intrinsically connected to transitive subgroups of the holomorph of a finite group, we present algorithms to classify and enumerate these objects for small degree, and apply them to obtain significant extensions to existing results. We also explore the classifications of these structures of degree $2pq$, where $p$ and $q$ are distinct odd primes. We conclude with some enumeration-inspired observations and a conjecture.