Exponents of Bogomolov multipliers
Primoz Moravec
本文证明了如果$G$是有限群，则在以下四种情况下，其Bogomolov乘子的指数整除$G$的指数：（i）$G$是亚阿贝尔群，（ii）$\exp G=4$，（iii）$G$是$\le 5$类的幂零群，（iv）$G$是$4$ -Engel群.
We prove that if $G$ is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of $G$ in the following four cases: (i) $G$ is metabelian, (ii) $\exp G=4$, (iii) $G$ is nilpotent of class $\le 5$, or (iv) $G$ is a $4$-Engel group.
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Computing Bogomolov multipliers and finitary Bogomolov multipliers of infinite groups
本文研究无限群的Bogomolov乘子，重点是多循环群，给出类二幂零群等结构描述，并提出两种在GAP中实现的算法。
In this paper, the Bogomolov multiplier of $p$-groups of order $p^7$ ($p>2$) and exponent $p$ is given.
The Bogomolov multiplier of a multiplicative Lie algebra
本文为乘法李代数引入Bogomolov乘子概念，建立Hopf型公式，并证明同斜乘法李代数的Bogomolov乘子同构。
We introduce the $q$-Bogomolov multiplier as a generalization of the Bogomolov multiplier, and we prove that it is invariant under $q$-isoclinism. We prove that the $q$-Schur Multiplier is invariant under $q$- exterior isoclinism, and as an easy consequence we prove that the Schur multiplier is invariant under exterior isoclinism. We also prove that if $G$, $H$ are $p$-groups and $G/Z^{\wedge}(G)\cong H/Z^{\wedge}(H)$, then the cardinality of the minimal number of generators of $G$ and $H$ are the same. Moreover we prove some structural results about $q$-nonabelian tensor square of groups.
The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is the subgroup of the Schur multiplier $H^2(G,\mathbb Q/\mathbb Z)$ consisting of the cohomology classes which vanish after restricting to every abelian subgroup of $G$. We give a new proof of a Hopf-type formula for $B_0(G)$ and derive an exact sequence for the cohomological version of the Bogomolov multiplier. Using this exact sequence we provide necessary and sufficient conditions for the corresponding inflation homomorphism to be an epimorphism and to be the zero map. Finally, we give a complete list of groups of order $p^6$, for odd…
Universal commutator relations, Bogomolov multipliers, and commuting probability
设$G$是有限$p$ -群.证明了当$G$的交换概率大于$(2p^2 + p - 2)/p^5$时，$G$ -不变函数域的非分歧Brauer群是平凡的.等价地，$G$中的所有映射之间的关系都是某些普适关系的结果。该界是最佳可能的，并给出了所有有限群的$1/4$的全局下界.这个结果是通过描述Bogomolov乘子非平凡且其所有真子群和真子群的Bogomolov乘子平凡的群的结构得到的.应用包括分类的$p$ -组的最低阶有非平凡的Bogomolov乘数和幂零类$2$，一个非概率标准消失的Bogomolov乘数，并建立一个序列的Bogomolov的绝对$γ$ -最小的因素，这是$2$ -组的任意大幂零类，从而提供反例的一些Bogomolov的索赔。在这方面，我们填补了空白的证明平凡的有限单群的Bogomolov乘子。
Local control and Bogomolov multipliers of finite groups
本文研究有限群 Bogomolov 乘子的局部控制问题，证明在 Sylow $p$-子群幂零类不超过 $p$ 时，其乘子的 $p$-部分具有局部受控性。
Bogomolov multipliers of all groups of order 128
本文是\cite{Mor11}与\cite{Eic08}结合给出的Bogomolov乘子算法的一个实现，以有效地确定$128$阶群的乘子。两个服务目的是结果\cite{Chu08,Chu09}的延续和应用程序\cite{Jez13}。
Bogomolov multiplier and the Lazard correspondence
本文将群的CP覆盖的概念推广到李代数的领域，证明了在群的情况下，李代数的所有CP覆盖都是同构的.最后证明了群和李环的CP覆盖在Lazard对应下也是Lazard对应的，并且Bogomolov乘子是同构的。
Bogomolov multipliers of $p$-groups of maximal class
设$G$是极大类和阶为$p^n$的$p$ -群.我们确定是否Bogomolov乘数$B_0(G)$是平凡的下中心序列的$G$和$P_1 = C_G(γ_2(G) / γ_4(G))$。如果此外$G$有积极程度的交换和$P_1$是亚阿贝尔，我们如何理解$B_0(G)$减少到更简单的换向器结构的$P_1$。这个结果覆盖了所有的$p$ -群的最大类的足够大的顺序，而且，它允许我们给的第一个自然家庭的$p$ -群包含丰富的群体与非平凡的Bogomolov乘子。我们还提供了更一般的结果，Bogomolov乘子的$p$ -群的任意余类$r$。
On some closure properties of the non-abelian tensor product and the Bogomolov multiplier
证明了有限幂零类、有限可解类、有限多环类、幂零类$n$的幂零类和超可解群在非交换张量积的构造下是封闭的.给出了有限生成群的非交换张量积为有限生成群的充要条件。证明了大多数有限单群的中心扩张都有平凡的Bogomolov乘子.
Units of group rings, the Bogomolov multiplier, and the fake degree conjecture
设$π$是有限$p$ -群，$\mathbb{F}_q$是有$q=p^n$元的有限域.用$\mathrm{I}_{\mathbb{F}_q}$表示群环$\mathbb{F}_q[π]$的增广理想。我们发现了$1+\mathrm{I}_{\mathbb{F}_q}$的阿贝尔化、$π$的Bogomolov乘子$\mathrm{B}_0(π)$和$π$的共轭类数$\mathrm{k}(π)$之间的惊人关系：\[ | (1+\mathrm{I}_{\mathbb{F}_q})_{\mathrm{ab}} |=q^{\mathrm{k}(π)-1}|\mathrm{B}_0(π)|. \]特别地，如果$π$是具有非平凡Bogomolov乘子的有限$p$ -群，则$1+\mathrm{I}_{\mathbb{F}_q}$是M.艾萨克
Bogomolov multipliers of word labelled oriented graph groups
本研究针对字标记有向图（WLOG）群，探讨其同调版本 Bogomolov 乘子 $\tilde{B_0}$，证明可通过底层 WLOG 计算生成元，将 Bestvina-Brady 群和 Artin 群归为 WLOG 群并计算两种乘子，为前者 Schur 乘子结构提供新证明。
The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial except for the case of certain covers of PSL(3,4).
Bogomolov multipliers for some $p$-groups of nilpotency class 2
有限群$G$的Bogomolov乘子$B_0(G)$被定义为Schur乘子的子群，该子群由限制到$G$的所有交换子群后消失的上同调类组成。Bogomolov乘数的平凡性是Noether问题的障碍。证明了若$G$是$G_1$和$K_i\leq Z(G_i), i=1,2$的中心积，且$θ:G_1\to G_2$是一个群同态，使得它的限制$θ\vert_{K_1}:K_1\to K_2$是一个同构，则$B_0(G_1/K_1), B_0(G_1)$和$B_0(G_2)$的平凡性蕴涵了$B_0(G)$的平凡性.我们给出了一个积极的答案Noether的问题，所有$2$ -发电机$p$ -群的幂零类$2$，并为一系列的$4$ -发电机$p$ -群的幂零类$2$（通常的要求为根的单位）。
The Bogomolov multiplier of rigid finite groups
有限群$G$的Bogomolov乘子被定义为Schur乘子的子群，该Schur乘子由限制到$G$的所有交换子群后消失的上同调类组成.这个不变量在商空间$G$的双有理几何中起着重要的作用。我们表明，在许多情况下，消失的Bogomolov乘数是保证刚性的$G$在这个意义上，它没有外部类保持自同构。
$\tilde{B_0}$-invariant of groups
该研究针对群簇推广了1988年提出、2012年被给出等价定义的Bogomolov乘子，得到了相关主题的若干新结果。
Unramified Brauer groups of finite and infinite groups
Bogomolov乘子是同构于给定商空间的非分歧Brauer群的群理论不变量。我们推导出一个同调版本的Bogomolov乘数，证明了一个霍普夫型公式，找到一个五项精确序列对应于这个不变量，并描述了中心扩展理论中的Bogomolov乘数的作用。给出了第二类幂零群的Bogomolov乘子的一个新刻画.我们定义的Bogomolov乘数在K-理论和证明其平凡性是等价于解决一个长期存在的问题所造成的低音。提出了一种计算Bogomolov乘子的算法。
Bogomolov multipliers for unitriangular groups
有限群$G$的Bogomolov乘子$B_0(G)$被定义为Schur乘子的子群，该子群由限制到$G$的所有交换子群后消失的上同调类组成。本文对Kang和Kunyavskiovskiy在\cite{KK}中提出的一个公开问题给出了肯定的回答.也就是说，我们证明了：如果$G$是$\f$上的单三角群，或者是它的下中心列的商，或者是它的下中心列的子群，或者是两个单三角群的中心积，则$B_0(G)=0$ .
Let $G$ be a finite group, $V$ a faithful finite-dimensional representation of $G$ over the complex field $\mathbb{C}$ and $\mathbb{C}(V)^{G}$ be the corresponding invariant field. The Bogomolov multiplier $B_{0}(G)$ of $G$ is canonically isomorphic to the unramified cohomological group $H_{\textrm{nr}}^{2}(\mathbb{C}(V)^{G},\mathbb{Q}/\mathbb{Z})$, which has been used by Saltman (1984) and Bogomolov (1988) to provide counter-examples to the rationality problem of $\mathbb{C}(V)^{P}$ for finite $p$-groups $P$ over $\mathbb{C}$. In this paper, we investigate the vanishing property of $B_{0}(P)$…
Topological perspectives on the vanishing of some Bogomolov multipliers
本文从拓扑视角出发，结合组合拓扑技术与Ihara-Yokonuma计算，证明有限单群和有限Coxeter群的博戈莫洛夫乘子均消失，为搭建其在代数几何与配边群中作用的桥梁提供了小步进展。
The Bogomolov multiplier of a group is the unramified Brauer group associated to the quotient variety of a faithful representation of the group. This object is an obstruction for the quotient variety to be stably rational. The purpose of this paper is to study these multipliers associated to nilpotent pro-$p$ groups by transporting them to their associated Lie algebras. Special focus is set on the case of $p$-adic Lie groups of nilpotency class $2$, where we analyse the moduli space. This is then applied to give information on asymptotic behaviour of multipliers of finite images of such groups…
Commutativity preserving extensions of groups
在平行的经典理论的中心扩展的群体，我们开发了一个版本的扩展，保持交换性。结果表明，Bogomolov乘子是一个泛对象parametrising这样的扩展的一个给定的组。最大和最小的扩展检查，并探索与通勤概率的连接。这些考虑产生了博戈莫洛夫乘数的指数和秩的界限。
Unramified Brauer groups and isoclinism
证明了如果$G_1$和$G_2$是等倾群，则它们的Bogomolov乘子同构.
On the Exponent of the Schur multiplier of a Pair of Finite $p$-Groups
当$N$在$G$中有补时，本文给出了有限$p$ -群对$(G,N)$的Schur乘子指数的一个上界.作为结果，我们表明，一对$(G,N)$的舒尔乘子的指数整除$\exp(N)$，如果$(G,N)$是一对有限的$p$ -群的类至多$p-1$。我们还证明了，如果$N$是强大的$G$嵌入，那么一对$(G,N)$的舒尔乘子的指数除$\exp(N)$。
In this note, we give short proofs of the well-known results that the exponent of the Schur multiplier $\M$ divides the exponent of $\G$ for finite $\p$-groups of maximal class and potent $\p$-groups. Moreover, we prove the same for a finite $\p$-group $\G$ satisfying $\G^{\p^2}\subset γ_{\p}(\G)$, and for $3$-groups of class $5$. We do this by proving a general lemma, and show that these three classes of groups satisfy the hypothesis of our lemma.
On the Exponent of the Schur Multiplier
一个长期存在的问题归因于我。舒尔说，对于有限群$G$，第二个同调群$H_2(G, \mathbb{Z})$的指数除以$G$的指数。本文对幂零类为5的奇指数有限幂零群、$p$ -中心亚交换$p$群以及L. E . \cite{LEW}中的Wilson。此外，我们还改进了一些作者给出的界限。
The unitary cover of a finite group and the exponent of the Schur multiplier
对于有限群，我们引入了一个特殊的中心扩张，酉覆盖，在那些满足投影提升性质的扩张中具有最小的指数。我们得到了与次正规列有关的Schur乘子的指数的新的界，并且发现了新的族，其界是群的指数。最后，我们证明了酉覆盖是由2-生成元群的限制性Burnside问题的Zel'manov解控制的。
There is a long-standing conjecture attributed to I Schur that if $G$ is a finite group with Schur multiplier $M(G)$ then the exponent of $M(G)$ divides the exponent of $G$. It is easy to see that this conjecture holds for exponent 2 and exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. In this note I give an example of a group $G$ with exponent 5 with Schur multiplier $M(G)$ of exponent 25, and an example of a group $A$ of exponent 9 with Schur multiplier $M(A)$ of exponent 27.
On a Conjecture of a Bound for the Exponent of the Schur Multiplier of a Finite $p$-Group
设$G$是幂零类$k$的一个$\exp(G)$-群，其指数为有限，令$m=\lfloor\log_pk\rfloor$ .我们证明了对于所有的$c\geq1$，$\exp(M^{(c)}(G))$整除$\exp(G)p^{m(k-1)}$，其中$M^{(c)}(G)$表示$G$的c-幂零乘子。这意味着$\exp(M(G))$对类至多为$p-1$的所有有限$p$ -群整除$\exp(G)$。此外，我们还证明了我们的结果是对M给出的$M^{(c)}(G)$的指数上界的改进。R. Jones，G.埃利斯和P.莫拉维克在某些情况下。
Invariant differential operators and central Fourier multipliers on exponential Lie groups
设$G$是指数可解李群。根据定义，$G$是$\ast$-正则的，如果对于$G$的所有酉表示$π$，$ker_{L^1(G)}π$在$ker_{C^\ast(G)}π$中是稠密的。Boidol用纯代数条件刻画了$\ast$-正则指数李群.在本文中，我们将重点关注非$\ast$-正则群。我们说$G$是本原$\ast$-正则的，如果对所有的不可约表示都满足上述密度条件.我们的目标是开发适当的工具来验证这种较弱的属性。为此，我们将介绍Duflo对$(W,p)$和中央傅立叶乘法器$ψ$上的稳定器$M=G_fN$的代表$π=\mcK(f)$在一般的位置。利用Littlewood-Paley理论，我们将得到一些关于乘子算子$T_ψ$的结果，这些结果可能是独立有趣的。通过详细研究两个重要的例子，我们将勾勒出分离三元组$(W,p,ψ)$的方法的范围。值得注意的是，"分离三元组"和"对子元的限制"方法足以证明所有维数为$\le 7$的指数可解李群都是本原$\ast$-正则的.
The exponents of $p$-groups of maximal class and their Schur multipliers
研究阶为\(p^n\)（\(p,n>2\)）的极大类\(p\)-群的指数及舒尔乘子，通过分析\(n\)与\(p\)的关系确定群指数，得出舒尔乘子指数至多为特定值，明确了极大类\(p\)-群相关指数的规律。
On p-central groups
通过引入低$p$-中心列的一个变体，将奇素数$p$的自由$p$ -中心群的概念推广到$p=2$的情形.这使我们能够计算自由$p$ -中心群的Schur乘子。证明了对任何$p$ -中心群，其Schur乘子的指数除群的指数，并确定其指数秩.
The ambiguity index of an equipped finite group
在\cite{Ku0}中，为每个装备的有限群$(G,O)$引入模糊度指数$a_{(G,O)}$。它等于用伽罗瓦群$G$表示的射影线的覆盖的Hurwitz空间的连通分支数，假定所有局部单值点都属于$G$中的共轭类$O$，且分支点的数目大于某个常数.在这篇文章中，我们证明了模糊度指数可以确定的大小的推广所谓的博戈莫洛夫乘数（\cite{Kun1}，也见\cite{BO87}），因此可以很容易地计算出许多对$(G,O)$。
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。
Primoz Moravec published a very important paper in 2007 where he proved that if $G$ is a finite group of exponent $n$ then the exponent of the Schur multiplier of $G$ can be bounded by a function $f(n)$ depending only on $n$. Moravec does not give a value for $f(n)$, but actually his proof shows that we can take $f(n)=ne$ where $e$ is the order of $b^{-n}a^{-n}(ab)^{n}$ in the Schur multiplier of $R(2,n)$. (Here$R(2,n)$ is the largest finite two generator group of exponent $n$, and we take $a,b$ to be the generators of $R(2,n)$.) It is an easy hand calculation to show that $e=n$ for $n=2,3$, a…
On the Order of Schur Multipliers of Finite Abelian p-Groups
设$G$是一个具有$|M(G)|=p^{\frac{n(n-1)}{2}-t},$的$p^{n}$阶的有限$p$ -群，其中$M(G)$是$G$的Schur乘子.是的Berkovich，X. Zhou，G. Ellis已经确定了$G$的结构，当$t=0,1,2,3$。在本文中，我们将给出交换$p$群$G$的一些结构，其中$S_2M(G)$是$G$的亚交换乘子，并满足$G, M(G),$和$S_2M(G)$的指数的条件.
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。
There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past. Let a group $G$ be presented as the quotient of a free group $F$ by a normal subgroup $R$. Given a positive integer $c$, the $c$-nilpotent multiplier of the group $G$ is the abelian group $\mathcal M^{(c)}(G)=(R\cap γ_{c+1}(F))/γ_{c+1}(R,F)$, where $γ_1(R,F)=R$, $γ_{c+1}(R,F)=[γ_c(R,F),F]$, and $γ_{c+1}(F)=γ_{c+1}(F,F)$. In particular, $\mathcal{M}^{(1)}(G)$ is the Schur multiplier of $G$. The crucial aspect of the research in to the $c$-nilpotent multipliers of groups includes either …
A note on some special $p$-groups
最近Rai得到了$d$ -生成元特殊$p$ -群的Schur乘子阶的一个上界，当它的导出子群对于$ d\geq 3 $和$ p\neq 2. $具有最大值$ p^{\frac{1}{2}d(d-1)}$时.这里我们试图得到这种$p$ -群的Schur乘子，外平方和张量平方.然后我们指定哪些人有能力。当$ |G'|=p^{\frac{1}{2}d(d-1)-1}$为$ d\geq 3 $和$ p\neq 2. $时，给出了$d$生成元群$p$的Schur乘子、外积和张量平方的阶的上界。另外，当$ G $为指数$ p, $时，给出了$ G. $的结构
Absence of twisting for non-trivial discrete torsion
本文从Dijkgraaf-Witten视角研究n- торus的离散扭力，引入untwisted类子群Br^n(G)，并推导出相关系数精确序列，同时计算了有限子群的分区函数。
Presenting a finite group by a free product of finite cyclic groups the Hopf formula for the Schur multiplier affords also a covering group, and this has minimal exponent provided that the order of the generators is preserved. This condition corresponds to a covering projection between simplicial complexes, and so a presentation by a Fuchsian group corresponds to a covering projection between compact surfaces.
On the size of the Schur multiplier of finite groups
本文改进了有限 p-群和有限群 Schur 乘子及第二上同调群大小的界，该界依赖于群的生成元数等参数，并在特定条件下构造了达到该界的 capable p-群。
Schur multiplier and Schur covers of relative Rota-Baxter groups
本文研究相对Rota-Baxter群，引入其Schur乘子与Schur覆盖，证明有限情形下指数整除性及Schur覆盖的弱同斜性，与斜左括号结果一致。
We establish precise regularity conditions for $L_p$-boundedness of Fourier multipliers in the group algebra of $SL_n(\mathbf{R})$. Our main result is inspired by Hörmander-Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity order. The asymptotics also match Mikhlin formula for a exponentially growing weight with respect to the word length. Additional decay comes imposed by this growth and Mikhlin condition for high order terms. Laff…
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