Hopf Galois structures on extensions of degree twice an odd prime square and their associated left braces

Skew Braces and Hopf-Galois Structures of Heisenberg Type

We classify all skew braces of Heisenberg type for a prime number $ p>3 $. Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew braces and Hopf-Galois theory, we can determine all Hopf-Galois structures of Heisenberg type on Galois field extensions of fields of degree $ p^{3} $.

无真左理想的斜辫子

Skew braces with no proper left ideals

本文研究有限左单斜辫子的部分分类，其中左单斜辫子定义为没有非平凡真左理想的斜辫子，并与有限域扩张上的极小Hopf-Galois结构一一对应。

Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group

Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete charac…

斜辫群胚

Skew bracoids

本文将斜辫推广为斜辫群胚，建立其分类与结构理论，并证明有限斜辫群胚对应于有限可分域扩张上的Hopf-Galois结构，从而推广了有限斜辫与Galois扩张的已知对应关系。

Commutator-central maps, brace blocks, and {H}opf-{G}alois structures on {G}alois extensions

Let $G$ be a nonabelian group. We show how a collection of compatible endomorphisms $ψ_i:G\to G$ such that $ψ_i([G,G])\le Z(G)$ for all $i$ allows us to construct a family of bi-skew braces called a brace block. We relate this construction to other brace block constructions and interpret our results in terms of Hopf-Galois structures on Galois extensions. We give special consideration to the case where $G$ is of nilpotency class two, and we provide several examples, including finding the maximal brace block containing the group of quaternions.

Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the cyclic Sylow $p$-subgroup case

$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $Γ$ are groups of order $p^{2} q$ with non-is…

Abelian maps, bi-skew braces, and opposite pairs of {H}opf-{G}alois structures

Let $G$ be a finite nonabelian group, and let $ψ:G\to G$ be a homomorphism with abelian image. We show how $ψ$ gives rise to two Hopf-Galois structures on a Galois extension $L/K$ with Galois group (isomorphic to) $G$; one of these structures generalizes the construction given by a ``fixed point free abelian endomorphism'' introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equat…

Opposite skew left braces and applications

Given a skew left brace $\mathfrak{B}$, we introduce the notion of an "opposite" skew left brace $\mathfrak{B}'$, which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked with both solutions to the Yang-Baxter Equation and Hopf-Galois structures on Galois field extensions. We show that the set-theoretic solution to the YBE given by $\mathfrak{B}'$ is the inverse to the solution given by $\mathfrak{B}$; this allows us to identify the group-like elements in the Hopf algebra providing the Hopf-Galois structure using …

二面体斜括号

Bidihedral skew braces

研究加法群和乘法群同构于二面体群\(D_{n}\)的斜括号，通过分类得到二面体伽罗瓦扩张上二面体型霍普夫 - 伽罗瓦结构分类的新证明。

诱导 brace 与 Hopf Galois 结构

Inducing braces and Hopf Galois structures

本文证明大小为 np 的左 brace 可由大小为 p 与 n 的左 brace 半直积构造，并由此给出计算 abel 型 Hopf Galois 结构数的公式，进而完全描述大小为 12p 的左 brace 及其 Hopf Galois 结构数。

From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures

The interplay between set-theoretic solutions of the Yang--Baxter equation of Mathematical Physics, skew braces, regular subgroups, and Hopf--Galois structures has spawned a considerable body of literature in recent years. In a recent paper, Alan Koch generalised a construction of Lindsay N.~Childs, showing how one can obtain bi-skew braces $(G, \cdot, \circ)$ from an endomorphism of a group $(G, \cdot)$ whose image is abelian. In this paper, we characterise the endomorphisms of a group $(G, \cdot)$ for which Koch's construction, and a variation on it, yield (bi-)skew braces. We show how the s…

作为可解扩张上Hopf-Galois结构的类型出现的非阿贝尔单群

Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension

本研究确定了可解扩张上作为Hopf-Galois结构类型的有限非阿贝尔单群，同时给出了乘法群可解的斜brace中作为加法群的这类单群的完整列表。

$p^{2} q$ 次扩张上的 Hopf-Galois 结构与 $p^{2} q$ 阶斜括号：初等阿贝尔 Sylow $p$-子群情形

Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the elementary abelian Sylow $p$-subgroup case

本文完成了当伽罗瓦群 Sylow $p$-子群为初等阿贝尔群时，$p^{2}q$ 次扩张的 Hopf-Galois 结构及相应斜括号的分类，通过伽马函数与全形正则子群的一一对应发展了更概念化的枚举方法。

包含斜 brace 的斜 bracoid

Skew bracoids containing a skew brace

本文证明一类斜 bracoid 与左消去半 brace 一一对应，并借此获得和研究集合论 Yang--Baxter 方程的解，同时给出大量实例。

Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,μ)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the o…

所有Hopf–Galois对应均为双射的类型的分类

Classification of the types for which every Hopf--Galois correspondence is bijective

本研究借鉴Hopf–Galois结构与skew brace关联的方法，分类了所有Galois扩张上类型为N的Hopf–Galois结构对应均为双射的群N，拓展了此前针对G-Galois扩张的相关分类结果。

On $\mathbb{Z}_N\rtimes\mathbb{Z}_2$-Hopf-Galois structures

Let $K/F$ be a finite Galois extension of fields with $Gal(K/F)=Γ$. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. Dihedral group is one particular example of semidirect product of $\mathbb{Z}_n$ and $\mathbb{Z}_2$. In this article we count the number of Hopf-Galois structures with Galois group $Γ$ of type $G$, where $Γ,G$ are groups of the form $\mathbb{Z}_N\rtimes_ϕ\mathbb{Z}_2$ when $N$ is odd with radical of $N$ being a Burnside number. As an application we also find the corresponding number of skew braces.

Hopf-Galois Realizability of $\mathbb{Z}_n\rtimes\mathbb{Z}_2$

Let $G$ and $N$ be finite groups of order $2n$ where $n$ is odd. We say the pair $(G,N)$ is Hopf-Galois realizable if $G$ is a regular subgroup of $\h(N)=N\rtimes\au(N)$. In this article we give necessary conditions on $G$ (similarly $N$) when $N$ (similarly $G$) is a group of the form $\mathbb{Z}_n\rtimes\mathbb{Z}_2$. Further we show that this condition is also sufficient if radical of $n$ is a Burnside number. This classifies all the skew braces which has the additive group (or the multiplicative group) to be isomorphic to $\mathbb{Z}_n\rtimes\mathbb{Z}_2$, in this case.

几乎经典的斜偏近群

Almost classical skew bracoids

本文研究了斜偏近群的两个子类，探讨其在 Hopf-Galois 理论中的应用，并基于此视角给出了新的构造，恢复了已有结果，同时检验了由此产生的 Yang-Baxter 方程的解。