On the structure of some one-generator nilpotent braces
Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin
该研究刻画了单元素生成的幂零左 brace 的结构，为理解此类代数对象提供了详细结果。
本文详细描述了由单个元素生成的一些幂零左 brace。
This article provides a detailed description of some nilpotent left braces generated by one element.
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In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite Fp-braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent Fp-braces generated by one element of cardinality p^4 is obtained, answering a question posed by Leandro Vendramin. It is also shown that the sum of a finite number of left nilpotent ideals in a left brace is a left…
On finitely generated left nilpotent braces
本文描述了类至多为2的有限生成左幂零括号，利用左幂零类至多为2则右幂零类至多为3的性质，并在该范畴中构造了自由对象。
On left nilpotent skew braces of class 2
本文研究满足 $B^3=0$ 的类2左幂零斜 braces，证明幂零型必中心幂零并给出右幂零类界，阿贝尔型时类至多为3且界最优。
We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces.
On Simply Connected Simple Lie Skew Braces with Nilpotent Multiplicative Group
证明乘法群幂零的单连通单李斜辫必须是一维且交换的，即若乘法群幂零且维数大于1，则李斜辫不是单的。
We classify nilpotent pre-Lie rings of cardinality $p^4$ and thereby braces of the same cardinality, for a sufficiently large prime $p$. It has been shown that nilpotent pre-Lie rings of cardinality $p^n$ correspond to strongly nilpotent braces of the same cardinality, for sufficiently large $p$. These braces are explicitly obtained from the corresponding pre-Lie rings by the construction of the group of flows. Not right nilpotent braces of cardinality $p^4$ have been classified, hence our results finish the classification of braces of cardinality $p^4$.
Classification of right nilpotent $\mathbb{F}_p$-braces of cardinality $p^5$
本文利用势为 $p^5$ 的幂零 $\mathbb{F}_p$-brace 与同阶幂零 pre-Lie 代数之间的联系，完成了对势为 $p^5$ 的右幂零 $\mathbb{F}_p$-brace 的全面分类。
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.
On the Sylow Theorem for Skew Braces
研究有限斜格的Sylow定理，证明当斜格为双侧、双斜、右nilpotent、λ-同态或超可解时成立，并探讨了更多专门情形，证明了广义Hall型定理。
Finite skew braces of square-free order and supersolubility
本文研究超可解斜 braces，证明其具有 Sylow 塔并刻画中心幂零理想与多重置换层性质，同时给出判定相关结构斜 brace 是否超可解的算法。
Central nilpotency of left skew braces and solutions of the Yang-Baxter equation
本文研究中心幂零斜辫，引入指标、好的Fitting理想和Frattini理想，用吸收多项式刻画理想交换子，解决开放问题，并举例说明中心幂零理想乘积不封闭及理想化子一般不存在。
The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces
本文研究有限斜括号的左幂零剩余，证明其刻画最大左幂零商，并利用群论方法探讨其补的存在性，给出阶18的反例，最后联系KSV可解性。
From braces to pre-Lie rings
本文构造与有限 brace 商相关的 pre-Lie 环，利用 powerful Lie 环研究自同构与交换元界，并建立 powerful brace 和左幂零 pre-Lie 环的对应。
A note on right-nil and strong-nil skew braces
本研究完全解答了arXiv:1806.01127的两个问题，证明有限阿贝尔型强幂零斜brace未必右幂零，但幂零型且满足$b\ast b=0$时成立，并通过例子说明结果最优。
We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X,r) which are poly-Z groups but not Engel groups. We also show that a finite solution of the Yang-Baxter equation can be embedded in a convenient way into a finite brace and into…
Free Skew Braces and Free Solutions of the Yang--Baxter Equation
证明Yang-Baxter方程的自由非退化双射解的结构斜辫子是自由的，并构造任意类的自由右幂零斜辫子，证明其可解词问题及有限解中的律对一切解成立。
Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
本文研究斜括号中不同幂零性概念的关系，提出零化子幂零斜括号作为幂零群的类比，并在该框架下推广了群论中的若干著名定理，应用于Yang-Baxter方程解的结构分析。
Bisoluble and Binilpotent Skew Braces of Finite Morley Rank
研究有限Morley秩的连通斜括号，证明其加法与乘法群的幂零性或可解性可推出括号本身的左幂零性或弱可解性。
Inverse Baer deformations and finite simple skew braces of nilpotent type
本文引入逆Baer形变，从阿贝尔型辫子构造出加法幂零类至多为二的斜辫子，并利用非对称积的简单左辫子，对每对满足条件的奇素数构造了无穷多个有限单斜辫子，其加法群为非阿贝尔幂零类二群。
Sylow theory and the nilpotency class of left nilpotent skew braces
本文证明有限左幂零斜辫的乘法群中的Sylow p-子群是Sylow p-子辫，且每个p-子辫包含于某个Sylow p-子辫中，从而去掉了可解性假设，并给出了左幂零类的上界。
From actions of an abelian group on itself to left braces
提出一种利用阿贝尔群对自身的作用构造右幂零类至多二的左括号的方法，并由此恢复自由右幂零单生成左括号的构造。
Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right $p$-nilpotent skew brace…
On left braces in which every subbrace is an ideal
该研究定义并刻画了所有子 brace 均为理想的 Dedekind 左 brace，证明有限情形的中心幂零性，并完整描述加法群初等阿贝尔的情形。
On left braces in which every subbrace is an ideal II
本文研究了Dedekind braces的性质，证明了其为阿贝尔的充分条件，并探讨了与杨-巴克斯方程解的关系。
Skew braces are algebraic structures related to the solutions of the set-theoretic quantum Yang-Baxter equation. We develop the central nilpotency theory for such algebraic structures in the sense of Freese-McKenzie \cite{comm} and we compare the universal algebraic notion of central nilpotency with the notion of right and left $*$-nilpotency developed in \cite{NilpotentType}.
Commuting probability of skew left braces
研究斜左 brace 的交换概率，证明其上限为 3/4，且 3/4 是唯一在 (5/8,1) 区间内出现的有理数，同时探讨了其性质与应用。
Powerful multiplicative groups do not force right nilpotence in finite braces
该研究构造了一类有限左 brace，其乘法群为强大 $p$-群但非右幂零，否定了 Shalev–Smoktunowicz 猜想，还得到了 Yang–Baxter 方程的相关集合论解。
On some connections between braces and pre-Lie rings outside of the context of Lazard's correspondence
本文研究Lazard对应之外的brace与pre-Lie环的联系，在特定条件下证明因子brace可由左幂零pre-Lie环的流群公式构造。
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。
Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions $(X,r)$ and show, for example, that this property is equivalent to the solution associated to the structure monoid $M…
On pre-Lie rings related to some non-Lazard braces
本文研究素数阶brace，在若干条件下证明其关联pre-Lie环，并给出由左幂零pre-Lie环构造商brace的公式及其应用。
Let p be a prime number. We give the explicit structure of 2- nilpotent multiplier for each finite 2-generator p-group of class two. Moreover, 2-capable groups in that class are characterized.
Let $G$ be the free two step nilpotent Lie group on three generators and let $L$ be a sub-Laplacian on it. We compute the spectral resolution of $L$ and prove that the operators arising from this decomposition enjoy a Tomas-Stein type estimate.
Enumerating finite class-2-nilpotent groups on 2 generators
通过给出g（n，2，2）n^{-s}的Dirichlet生成函数\sum_{n=1}^\的显式公式，计算了由至多2个生成元生成的至多2类n阶幂零群的个数g（n，2，2）.
The normal zeta function of the free class two nilpotent group on four generators
我们明确地计算了四个生成元上自由类二幂零群的正规zeta函数。这是赫希长度十的
A maximal abelian normal subgroup A in a nilpotent group N is self-centralizing. This makes their role an important one in determining the structure of the nilpotent group. For example if A is finite then N is also finite. In the free nilpotent group of class 2 such a group is always a cyclic extension of the center. However this need not be true in an arbitrary finitely generated class two nilpotent group. We show that three generator nilpotent groups violating this condition can be easily found. However in the case when there are four or more generators certain conditions must hold.
The authors extend to the $q-$tensor square $G \otimes^q G$ of a group $G$, $q$ a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square $G \otimes G$ ($q = 0$). The results are applied to the computation of $G \otimes^q G$ for finitely generated nilpotent groups $G$, specially for free nilpotent groups of finite rank. They also generalize to all $q \geq 0$ results of M. Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square $G \otimes G$ when $G$ is a $n-$generator nilpo…
Normal subgroup growth in free class-2-nilpotent groups
设$F_{2,d}$表示$d$生成元上的自由2类幂零群.我们计算了正规zeta函数$ζ^\triangleleft_{F_{2,d}}(s)$，证明了它们满足局部函数方程，并确定了它们的收敛横坐标和极点阶。
Random walks on nilpotent groups driven by measures supported on powers of generators
本文研究了广义幂零群上测度的大族$μ_{S,a}$的卷积幂的衰减。这里，$S=(s_1,...,s_k)$是群元素的生成$k$元组，而$a= (α_1,...,α_k)$是区间$(0,2)$中实数的$k$元组.对称度量$μ_{S,a}$由$S^*=\{s_i^{m}, 1\le i\le k,\,m\in \mathbb Z\}$支持，并给出与$$(1+m)^{-α_i-1}$$到$s_i^{\pm m}$，$i=1,...,k,$ $m\in \mathbb N$成比例的概率。当$n$趋于无穷大时，我们确定了返回概率$μ_{S,a}^{(n)}(e)$的行为。这种行为以某种微妙的方式取决于$k$元组$a$和生成元$s_i$在较低中心序列$G_{j}=[G_{j-1},G]$ $G_1=G$中的位置之间的相互作用。
On finite groups whose Sylow subgroups have a bounded number of generators
设G是有限非幂零群，使得G的每个Sylow子群至多由d个元生成，且p是除|G|的最大素数.我们证明了G有一个非幂零像G/N，使得N是特征的，且其指数由一个函数d和p有界.这个结果将被用来证明G的弗拉蒂尼子群的指数是有界的.对于可解群，我们将给出明确的上界.
Some Inequalities for Nilpotent Multipliers of Powerful $p$-Groups
本文给出了幂p-群的c-幂零乘子（关于至多为$c \geq 1$类的幂零群簇的Baer不变量）的阶、指数和生成元个数的不等式.我们的结果推广了Lubotzky和Mann $^{,}$的一些结果（Journal of Algebra，105（1987），484-505.）到幂零乘数同时，我们给出了一些具体的例子，证明了所得结果的严密性，改进了已有的一些不等式。
Some Inequalities for Nilpotent Multipliers of Finite Groups
本文给出了有限群，特别是有限$p$ -群的$c$ -幂零乘子（关于至多为$c \geq 1$类的幂零群簇的Baer不变量）的阶、指数和生成元个数的不等式.我们的结果推广了M. R.琼斯和核磁共振莫卧儿此外，我们表明，我们的结果改进了以前的一些不等式。
The free nilpotent group $G_{m,n}$ of class $m$ and rank $n$ is the free object on $n$ generators in the category of nilpotent groups of class at most $m$. We show that $G_{m,n}$ can be recovered from its reduced group $C^*$-algebra, in the sense that if $H$ is any group such that $C^*_r(H)$ is isomorphic to $C^*_r(G_{m,n})$, then $H$ must be isomorphic to $G_{m,n}$.
It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is red…
Cutoff for random Cayley graphs of nilpotent groups
研究秩和幂零类有界的有限幂零群随机Cayley图上简单随机游走的截止现象，证明随机生成元情形下高概率出现截止，并给出一般生成元集合下谱隙和混合时间与阿贝尔化投影渐近等价的判据。
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphi…
On some relatively free pro-p groups
本文证明了在2个生成元上的相对自由中心-by-metabelian pro-p群与（2类幂零）-by-abelian pro-p群在所有profinite群类中均有限可公理化。
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