Hopf-Galois structures on finite extensions with quasisimple Galois group

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。

Hopf Galois structures on symmetric and alternating extensions

By using our previous results on induced Hopf Galois structures and a recent result by Koch, Kohl, Truman and Underwood on normality, we determine which types of Hopf Galois structures occur on Galois extensions with Galois group isomorphic to alternating or symmetric groups.

From Galois to Hopf Galois: theory and practice

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations and explicit descriptions of all the ingredients involved in a Hopf Galois structure. At the end we give just a glimpse of how this theory is used in the context of Galois module theory for wildly ramified…

On the connection between Hopf--Galois structures and skew braces

We present a different version of the well-known connection between Hopf--Galois structures and skew braces, building on a recent paper of A. Koch and P. J. Truman. We show that the known results that involve this connection easily carry over to this new perspective, and that new ones naturally appear. As an application, we present new insights on the study of the surjectivity of the Hopf--Galois correspondence, explaining in more detail the role of bi-skew braces in Hopf--Galois theory.

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} $.

Skew left braces and isomorphism problems for Hopf-Galois structures on Galois extensions

Given a finite group $ G $, we study certain regular subgroups of the group of permutations of $ G $, which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to $ G $ and Hopf-Galois structures admitted by a Galois extension of fields with Galois group isomorphic to $ G $. We study the questions of when two such subgroups yield isomorphic skew left braces or Hopf-Galois structures involving isomorphic Hopf algebras. In particular, we show that in some cases the isomorphism class of the Hopf algebra giving a Hopf-Galois…

Induced Hopf Galois structures

For a finite Galois extension K/k and an intermediate field F such that Gal(K/F) has a normal complement in Gal(K/k), we construct and characterize Hopf Galois structures on K/k which are induced by a pair of Hopf Galois structures on K/F and F/k.

关于四元数与二面体brace的数量及Hopf--Galois结构

On the number of quaternion and dihedral braces and Hopf--Galois structures

本研究证明了乘群为广义四元数群的给定阶braces数量的相关猜想，给出了二面体群情形的类似结果，并枚举了对应Galois扩张上的阿贝尔型Hopf-Galois结构。

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.

次数为 $pq$ 的可分域扩张上的 Hopf-Galois 结构

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

本文研究次数为 $pq$（$p,q$ 为不同奇素数）的可分但不一定正规的域扩张，利用其 Galois 闭包的两个 Galois 群刻画并计数其上的 Hopf-Galois 结构。

An algorithm to determine Hopf Galois structures

A Hopf Galois structure on a finite field extension L/K is given by a finite cocommutative K-Hopf algebra and a Hopf action. In this paper we present an algorithm 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 describe the results obtained for extensions of degree up to 11. Besides, we prove that separable extensions of degree equal to the square of an odd prime have at most one type of Hopf Galois structures.

Non-isomorphic Hopf-Galois structures with isomorphic underlying Hopf algebras

We give a degree 8 separable extension having two non-isomorphic Hopf-Galois structures with isomorphic underlying Hopf algebras.

Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$

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 degree up to eleven and several properties of those. Besides, we exhibit several results on Hopf Galois structures inspired by the program output. We prove that if $(H,μ)$ is an almost classically Hopf Galois structure, then it is the unique Hopf Galois structure with underlying Hopf algeb…

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…

Automatic realization of Hopf Galois structures

We consider Hopf Galois structures on a separable field extension $L/K$ of degree $p^n$, for $p$ an odd prime number, $n\geq 3$. For $p > n$, we prove that $L/K$ has at most one abelian type of Hopf Galois structures. For a nonabelian group $N$ of order $p^n$, with commutator subgroup of order $p$, we prove that if $L/K$ has a Hopf Galois structure of type $N$, then it has a Hopf Galois structure of type $A$, where $A$ is an abelian group of order $p^n$ and having the same number of elements of order $p^m$ as $N$, for $1\leq m \leq n$.

Hopf Galois structures on separable field extensions of odd prime power degree

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(\mathcal{H},μ)$, where $\mathcal{H}$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper, we present several results on Hopf Galois structures on odd prime power degree separable field extensions. We prove that if a separable field extension of odd prime power degree has a Hopf Galois structure of cyclic type, then it has no structure of noncyclic type. We determine the number of Hopf Galois structures of cyclic type on a separable field extension of degree $p^n$, $p$ an odd prime, such that the G…

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 …

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

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

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。

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。

关于Hopf-Galois理论中出现的斜撑（skew brace）的若干半直积

On some semidirect products of skew braces arising in Hopf-Galois theory

本文分类了作为理想与左理想半直积的斜撑，并据此分类Galois群为半直积的域扩张上由正规Hopf子代数和Hopf子代数实现的Hopf-Galois结构，进而研究广义正规基生成元与局部域扩张的整模结构。

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…

Abelian Hopf Galois structures from almost trivial commutative nilpotent algebras

Let $L/K$ be a Galois extension of fields with Galois group $G$, an elementary abelian $p$-group of rank $n$ for $p$ an odd prime. It is known that nilpotent $\mathbb{F}_p$-algebra structures $A$ on $G$ yield regular subgroups of the holomorph of $G$, hence Hopf Galois structures on $L/K$. In this paper we illustrate the richness of Hopf Galois structures on $L/K$ by examining the case where $A$ is abelian of dimension $n$ where the dimension of $A^2 = 1$. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent …

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…

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…

循环类型上的平行扩展的Hopf-Galois结构

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

本文研究了当扩展L/K的度数为素数幂且类型为循环时，平行扩展是否也具有相同Hopf-Galois结构的问题，采用群论方法并结合Greither-Pareigis和Byott的工作予以解决。

Enumerating Dihedral Hopf-Galois Structures Acting on Dihedral Extensions

The work of Greither and Pareigis details the enumeration of the Hopf-Galois structures (if any) on a given separable field extension. For an extension $L/K$ which is classically Galois with $G=Gal(L/K)$ the Hopf algebras in question are of the form $(L[N])^{G}$ where $N\leq B=Perm(G)$ is a regular subgroup that is normalized by the left regular representation $λ(G)\leq B$. We consider the case where both $G$ and $N$ are isomorphic to a dihedral group $D_n$ for any $n\geq 3$. Using the normal block systems inherent to the left regular representation of each $D_n$,(and every other regular permu…

平行扩张上的 Hopf-Galois 结构

Hopf-Galois structures on parallel extensions

本文研究有限可分域扩张及其平行扩张上 Hopf-Galois 结构的关系，发现存在具有该结构但平行扩张缺失的现象，并证明可扩展为无限族，同时在奇素数次扩张中证明无此现象。

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…

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.

构造小次数的Hopf- Galois结构和斜bracoids

Constructing Hopf-Galois structures and skew bracoids of small degree

本文提出算法分类和枚举小次数的Hopf-Galois结构和斜bracoids，扩展了现有结果，并探讨了2pq次结构的分类，最后提出一些观察和猜想。

Characteristic Subgroup Lattices and Hopf-Galois Structures

The Hopf-Galois structures on normal extensions $K/k$ with $G=Gal(K/k)$ are in one-to-one correspondence with the set of regular subgroups $N\leq B=Perm(G)$ that are normalized by the left regular representation $λ(G)\leq B$. Each such $N$ corresponds to a Hopf algebra $H_N=(K[N])^G$ that acts on $K/k$. Such regular subgroups $N$ need not be isomorphic to $G$ but must have the same order. One can subdivide the totality of all such $N$ into collections $R(G,[M])$ which is the set of those regular $N$ normalized by $λ(G)$ and isomorphic to a given abstract group $M$ where $|M|=|G|$. There arises…

Hopf Galois结构，阶为$p^nq$的群的斜brace：循环Sylow子群情形

Hopf Galois structures, skew braces for groups of size $p^nq$: The cyclic Sylow subgroup case

本文研究了阶为$p^nq$且Sylow-$p$-子群为循环群的群上的Hopf-Galois结构，枚举了Galois $G$-扩张上类型为$N$的结构，计算了对应的斜brace数量，并在$q<p$时给出了完整分类。

与Cunningham链相关的可分域扩张上的Hopf-Galois结构

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

本文研究平方自由度数且素数满足Cunningham链关系的可分域扩张上的Hopf-Galois结构，扩展了Byott和Martin-Lyons的结果。

Some Braces of Cardinality $p^{4}$ and Related Hopf-Galois Extensions

We describe all $\mathbb F_{p}$-braces of cardinality $p^{4}$ which are not right nilpotent. Our solution illustrates a general way of investigating $\mathbb F_{p}$-braces of cardinality $p^{n}$ with a given multiplicative group. The constructed braces are left nilpotent, solvable and prime, and they also contain a non-zero strongly nilpotent ideal.

Hopf-Galois Structures Arising From Groups with Unique Subgroup of Order p

For $Γ$ a group of order $mp$ for $p$ prime where $gcd(p,m)=1$, we consider those regular subgroups $N\leq Perm(Γ)$ normalized by $λ(Γ)$, the left regular representation of $Γ$. These subgroups are in one-to-one correspondence with the Hopf-Galois structures on separable field extensions $L/K$ with $Γ=Gal(L/K)$. This is a follow up to the author's earlier work where, by assuming $p>m$, one has that all such $N$ lie within the normalizer of the $p$-Sylow subgroup of $λ(Γ)$. Here we show that one only need assume that all groups of a given order $mp$ have a unique $p$-Sylow subgroup, and that $p…

$p^{2} q$ 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。

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。