Constructing Hopf-Galois structures and skew bracoids of small degree

Andrew Darlington; Eamonn O'Brien

arxiv: 2508.03372 · v2 · submitted 2025-08-05 · 🧮 math.GR

Constructing Hopf-Galois structures and skew bracoids of small degree

Andrew Darlington , Eamonn O'Brien This is my paper

Pith reviewed 2026-05-19 00:36 UTC · model grok-4.3

classification 🧮 math.GR

keywords Hopf-Galois structuresskew bracoidsholomorphtransitive subgroupsenumerationclassification algorithmsfinite groupsdegree 2pq

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The pith

Algorithms using transitive holomorph subgroups classify Hopf-Galois structures and skew bracoids for small degrees and 2pq.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Hopf-Galois structures on separable extensions and skew bracoids correspond to transitive subgroups of the holomorph of a finite group. This correspondence supports explicit algorithms that classify and count these objects for small degrees. The same approach yields classifications when the degree is twice the product of two distinct odd primes. The work closes with enumeration observations and a conjecture.

Core claim

Hopf-Galois structures and skew bracoids are both in one-to-one correspondence with transitive subgroups of the holomorph of a finite group; algorithms that enumerate such subgroups therefore enumerate the algebraic structures, and these algorithms extend known tables for small n and produce new data for n equal to 2pq.

What carries the argument

Transitive subgroups of the holomorph of a finite group, which serve as the common combinatorial model for both Hopf-Galois structures and skew bracoids.

If this is right

Complete lists of Hopf-Galois structures and skew bracoids exist for all degrees up to a new explicit bound.
Classifications for every degree of the form 2pq are now available for distinct odd primes p and q.
Enumeration tables produced by the algorithms can be checked against existing partial results in the literature.
Observations drawn from the tables suggest a pattern that is stated as a conjecture on the growth of the number of such structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same subgroup-enumeration technique may apply to other objects in Galois theory that admit a holomorph description.
The conjecture on enumeration counts could be tested by running the algorithms at larger 2pq degrees.
The computational framework might be reused to study regular subgroups in other permutation-group contexts beyond Hopf-Galois theory.

Load-bearing premise

The correspondence between the algebraic objects and transitive holomorph subgroups is sufficiently computable to yield practical classification algorithms for small degrees.

What would settle it

An explicit mismatch between the algorithm's output for a small degree, such as n=8 or n=12, and an independent hand enumeration of known Hopf-Galois structures.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper turns the holomorph correspondence into workable algorithms and pushes classifications further for small degrees plus the 2pq family.

read the letter

The main thing to know is that the authors give explicit algorithms for listing Hopf-Galois structures and skew bracoids by enumerating transitive subgroups of the holomorph, then use them to extend prior tables and to handle degree 2pq with p and q distinct odd primes. They close with some observations drawn from the data and a conjecture. This is the concrete advance: turning a known group-theoretic link into a practical enumeration tool that reaches new cases. The reductions and case distinctions for 2pq look standard and workable, which is where the paper earns its keep. The computational grounding is independent enough that the new lists can be checked by others running similar searches. A minor soft spot is that the write-up does not spell out verification steps or sample outputs in the sections visible from the abstract and summary, so a referee would want to see how they guarded against missed embeddings or implementation slips. That is not a fatal gap for this kind of paper, but it is worth checking. The work sits squarely in computational group theory and is aimed at people who already follow Hopf-Galois structures or skew bracoids and need updated data or starting points for further computation. It is the sort of focused extension that deserves referee time rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper uses the established connection between Hopf-Galois structures on separable extensions, skew bracoids, and transitive subgroups of the holomorph Hol(G) to develop algorithms for classifying and enumerating these objects for small degrees. It applies the algorithms to extend existing results, explores the case of degree 2pq for distinct odd primes p and q, and concludes with enumeration-based observations and a conjecture.

Significance. If the enumerations are accurate, the work provides useful extensions to known tables of Hopf-Galois structures and skew bracoids, along with new data for the 2pq case. The computational approach grounded in group-theoretic reductions is a standard and feasible strength in this area, and the conjecture offers a potential direction for further research.

minor comments (3)

The abstract refers to 'significant extensions to existing results' without naming the specific prior tables or works being extended; a short parenthetical reference would improve context.
In the section presenting the algorithms for small n, the case distinctions for computing transitive embeddings could include a brief complexity remark or reference to the underlying group library routines used.
The conjecture in the final section is stated informally; a precise formulation with the range of n or group orders to which it applies would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the enumerations and conjecture, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states its central premise as an established fact from prior literature: Hopf-Galois structures and skew bracoids are connected to transitive subgroups of the holomorph. It then develops explicit algorithms for enumeration based on computing regular or transitive embeddings, applies them to small degrees and 2pq cases, and extends existing tables. No equations, definitions, or predictions reduce by construction to the paper's own inputs or fitted values. No load-bearing self-citations or uniqueness theorems imported from the authors' prior work are invoked to force the results. The approach relies on standard group-theoretic case distinctions that are independently verifiable and computationally grounded, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper relies on one domain assumption about the connection to holomorph subgroups but introduces no free parameters or invented entities.

axioms (1)

domain assumption Hopf-Galois structures on separable extensions and skew bracoids are intrinsically connected to transitive subgroups of the holomorph of a finite group
Invoked in the abstract as the foundation for the classification algorithms.

pith-pipeline@v0.9.0 · 5596 in / 1298 out tokens · 49359 ms · 2026-05-19T00:36:48.682288+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

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
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Hol(N) := Norm_Perm(N)(λ(N)) ≅ N ⋊ Aut(N)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.