arXiv subjects

Andrew Darlington

Publications and source records attributed to Andrew Darlington.

6 recordsLinked to original sources

arXiv2026

Indecomposable solutions with permutation brace of size $p^3$ and permutation braces of small sizes

Using the construction of Bachiller, Ced\'o and Jespers, we give a complete classification of the indecomposable involutive set-theoretic solutions to the Yang-Baxter equation whose permutation brace has size $p^3$, where p is an odd prime. We also give an algorithm for systematically producing all indecomposable involutive solutions with a given permutation brace, and use this to enumerate these with the permutation brace of size up to 107.

math.GR↗

arXiv2025

Hopf--Galois structures of cyclic type on parallel extensions of prime power degree

Let $L/K$ be any finite separable extension with normal closure $\widetilde{L}/K$. An extension $L'/K$ is said to be $\textit{parallel to $L/K$}$ if $L'$ is an intermediate field of $\widetilde{L}/K$ with $[L':K]=[L:K]$. We study the following question -- Given that $L/K$ admits a Hopf--Galois structure of type $N$, does it imply that every extension parallel to $L/K$ also admits a Hopf--Galois structure of type $N$? We completely solve this problem when the degree $[L:K]$ is a prime power and the type $N$ is cyclic. Our approach is group-theoretic and uses the work of Greither--Pareigis and Byott.

math.GR↗

arXiv2025

Constructing Hopf-Galois structures and skew bracoids of small degree

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.

math.GR↗

arXiv2025

Hopf-Galois structures on separable field extensions of degree related to Cunningham chains

The past few years have seen Hopf--Galois structures on extensions of squarefree degree studied in various contexts. The Galois case was fully explored by Alabdali and Byott in 2020, followed by a first attempt at generalising these results to include non-normal extensions by Byott and Martin-Lyons; their work looks at separable extensions of degree $pq$ with $p,q$ distinct odd primes, and $p=2q+1$. This paper extends the latter work further by considering separable extensions of squarefree degree $n=p_1...p_m$ where each pair of consecutive primes $p_i,p_{i+1}$ are related by $p_i=2p_{i+1}+1$.

math.GR↗

arXiv2025

Hopf-Galois structures on parallel extensions

Let $L/K$ be a finite separable extension of fields of degree $n$, and let $E/K$ be its Galois closure. Greither and Pareigis showed how to find all Hopf--Galois structures on $L/K$. We will call a subextension $L'/K$ of $E/K$ \textit{parallel} to $L/K$ if $[L':K]=n$. In this paper, we investigate the relationship between the Hopf--Galois structures on an extension $L/K$ and those on the related parallel extensions. We give an example of a transitive subgroup corresponding to an extension admitting a Hopf--Galois structure but that has a parallel extension admitting no Hopf--Galois structures. We show that once one has such a situation, it can be extended into an infinite family of transitive subgroups admitting this phenomenon. We also investigate this fully in the case of extensions of degree $pq$ with $p,q$ distinct odd primes, and show that there is no example of such an extension admitting the phenomenon.

math.GR↗

arXiv2024

Hopf-Galois structures on separable field extensions of degree $pq$

In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions $L/K$ is a natural next step. One must consider now the interplay between two Galois groups $G=\operatorname{Gal}(E/K)$ and $G'=\operatorname{Gal}(E/L)$, where $E$ is the Galois closure of $L/K$. In this paper, we give a characterisation and enumeration of the Hopf-Galois structures arising on separable extensions of degree $pq$ where $p$ and $q$ are distinct odd primes. This work includes the results of Byott and Martin-Lyons who do likewise for the special case that $p=2q+1$.

math.GR↗