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Constructing Hopf-Galois structures and skew bracoids of small degree

Andrew Darlington, Eamonn O'Brien

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.

Constructing Hopf-Galois structures and skew bracoids of small degree

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 , where and are distinct odd primes. We conclude with some enumeration-inspired observations and a conjecture.

Paper Structure

This paper contains 20 sections, 9 theorems, 51 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transitive subgroups of the holomorph
  4. Hopf--Galois structures
  5. Skew bracoids
  6. The Algorithms
  7. Recovering the action on $N$
  8. The action for Hopf--Galois structures
  9. The action for skew bracoids
  10. Realising Step 2
  11. Realising Step 3
  12. Exploiting parallelisation
  13. Enumerations and structures
  14. Enumerations
  15. Structures
  16. ...and 5 more sections

Key Result

Proposition 2.3

Let $M_1$ and $M_2$ be subgroups of $\mathrm{Hol}(N)$ such that $M_1$ is transitive on $N$. Then $M_1$ is conjugate to $M_2$ in $\mathrm{Hol}(N)$ if and only if they are conjugate by an element of $\mathrm{Aut}(N)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Summary of results for degree $2pq$

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 2.5: Byott's translation
  • Lemma 2.6: Byo96
  • Remark 2.7
  • Proposition 2.8
  • ...and 12 more