Powers in finite groups
Dan Segal, Nikolay Nikolov
本文证明了：若$G$是一个可生成的投射有限群，则其动词子群$G^{q}$是开的.等价地，在一个$d$生成元有限群中，每个$q$的次幂积都是$f(d,q)$的$q$次幂积.
In this note we prove that if $G$ is a finitely generated profinite group then the verbal subgroup $G^{q}$ is open. Equivalently in a $d$-generator finite group every product of $q$th powers is a product of $f(d,q)$ $q$th powers.
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We show that residually finite by residually finite extensions are weakly sofic.
The Fundamental group of a finite graph of conjugacy separable groups with finite edge groups is conjugacy separable
本文证明了具有有限边群的共轭可分群的有限图的基本群是共轭可分的，肯定回答了Ashot Minasyan提出的自然问题。
Finite abelian groups are Schur multipliers of finite solvable groups
证明每个有限阿贝尔群均可作为有限可解群的 Schur 乘子，进而推出其为某数域扩张的结群。
On the outer automorphism groups of finitely generated, residually finite groups
布马金-怀斯提出的问题，是否每一个可数群可以实现为外自同构群的一个非线性生成，剩余有限群。我们给出了一个部分的答案，这个问题的递归可呈现的群体。
On the conjugacy separability in the class of finite $p$-groups of finitely generated nilpotent groups
证明了对任意素数$p$，一个生成的幂零群在有限$p$ -群类中共轭可分当且仅当它的扭子群是有限$p$ -群且扭子群的商群是阿贝尔群.
Fixed points of finite groups acting on generalised Thompson groups
本文研究了Thompson群F的推广的有穷级自同构的中心化子和这些群的有限扩张中有限子群的共轭类。特别是，我们表明，在这些广义汤普森群的有限自同构的中心化要么承认一个有限型分类空间或不生成的。作为应用，我们推出了这类有限扩张的Bredon型结果.
Automorphism group of the reduced power (di)graph of a finite group
本文描述并计算了有限群的有向与无向约化幂图的完整自同构群，并建立了其与幂图自同构群之间的关系。
The power graph of a group is the graph whose vertex set is the set of nontrivial elements of group, two elements being adjacent if one is a power of the other. We introduce some way for find the automorphism groups of some graphs. As an application We describe the full automorphism group of the power graph of all finite groups. Also we obtain the full automorphism group of power graph of abelian, homocyclic and nilpotent groups
Embedding free Burnside groups in finitely presented groups
本文构造了奇$n > 2^{48}$的自由Burnside群$B(m,n)$的嵌入，并证明了秩为$m >1$的自由Burnside群$B(m,n)$在具有特殊性质的可表示群中的嵌入.这个嵌入的主要应用是一个简单的结构，不含非循环自由子群（这为顺从群上的冯·诺伊曼问题提供了一个简单的反例）。作为另一个应用，我们构造了奇指数$n \gg 1$的弱可表示群，它们不是局部有限的。
We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also valid for infinite graphs of groups.
Finiteness properties of profinite groups
广义地说，群的有限性是具有有穷级的性质的任何推广。无限群理论的很大一部分是关于有限性以及它们之间的关系。投射有限群是这方面的一个重要例子，它是紧拓扑群，与它们的有限象有密切的联系。本文研究了profinite群可能具有的几个有限性之间的关系，以及有限群和投射有限群的结构。
A Survey on Automorphism Groups of Finite $p$-Groups
有限p-群的自同构群的研究主要集中在三个方面：对常见的有限p-群，如Chevalley群的特殊p-群和Sylow p-子群的显式计算;构造具有特定自同构群的p-群;发现其自同构群本身是或不是p-群的有限p-群。该材料以不同的详细程度呈现，其中一些示例给出了完整的细节。
Groups that can be approximated by finite groups have been the center of much research. This has led to the investigations of the subgroups of metric ultraproducts of finite groups. This paper attempts to study the dual problem: what are the quotients of ultraproducts of finite groups? Since an ultraproduct is an abstract quotient of the direct product, this also led to a more general question: what are the abstract quotients of profinite groups? Certain cases were already well-studied. For example, if we have a pro-solvable group, then it is well-known that any finite abstract quotients are s…
Embedding finitely presented self-similar groups into finitely presented simple groups
该研究证明有限表现自相似群可嵌入有限表现单群，确立这类群满足Boone-Higman猜想，还通过环上矩阵群例子证明$\n\b{GL}_n(\n\b{Q})$的有限生成子群也满足该猜想。
On controllers of prime ideals in group algebras of torsion-free abelian groups of finite rank
本文考虑了有限秩阿贝尔群的群代数中素理想的一些研究方法
On finite generation of self-similar groups of finite type
一个有限型自相似群是正则有根树的所有自同构的投射有限群，这些自同构局部围绕每个顶点充当给定的允许作用的有限群群的元素。我们提供的标准，以确定当一个自相似组的有限型是有限的，水平传递，或拓扑生成。使用这些标准和GAP计算，我们表明，对于二进制字母表有没有无限的拓扑生成的自相似组的模式的深度$3$，并有$32$这样的群体的深度$4$。
A tubular group $G$ is a finite graph of groups with $\mathbb{Z}^2$ vertex groups and $\mathbb{Z}$ edge groups. We characterize residually finite tubular groups: $G$ is residually finite if and only if its edge groups are separable. Methods are provided to determine if $G$ is residually finite. When $G$ has a single vertex group an algorithm is given to determine residual finiteness.
On residual finiteness of graphs of free groups with cyclic edge groups
本研究刻画了边群为循环群的自由群有限图对应的群的剩余有限性，给出等价判定条件并通过构造有限标记图将其转化为图的易检测性质，证明了Wise的相关猜想。
Hydra group doubles are not residually finite
2013年，Kharlampovich、Myasnikov和Sapir构造了第一个具有大Dehn函数的剩余有限群的例子。给定任意递归函数$f$，它们产生一个Dehn函数控制$f$的可表示剩余有限群.关于具有超指数Dehn函数的剩余有限群，还没有已知的初等例子. Dison和Riley的Hydra群可以用来构造一个群序列，使得第k个群的Dehn函数等价于第k个Ackermann函数。Kharlampovich，Myasnikov和Sapir问是否这些群体是剩余有限的。我们表明，这些建设不产生剩余有限群。
Finiteness results for subgroups of finite extensions
本文在有限扩张的背景下讨论了Takahasi和Howson关于自由群子群的两个经典定理。我们提供了范围内的子群的交集类的群体，如虚拟自由群，虚拟幂零群或基本群的有限图形的群体与虚拟多环顶点群和有限棱道群的秩。作为Takahasi定理推广的一个应用，我们给出了有限群图的基本群的任意自同态的周期子群的秩的一个一致界，该图的群具有n-生成的虚幂零顶点群和有限棱道群.
In this paper we describe all the finite almost simple groups whose Gruenberg--Kegel graphs coincide with Gruenberg--Kegel graphs of finite solvable groups.
A slice (G, S) of finite groups is a pair consisting of a finite group G and a subgroup S of G. In this paper, we show that some properties of finite groups extend to slices of finite groups. In particular, by analogy with B-groups, we introduce the notion of T$^\circ$-slice, and show that any slice of finite groups admits a largest quotient T$^\circ$-slice.
Embeddings of relatively free groups into finitely presented groups
我们构造了相对自由群（比如自由Burnside群、自由可解群）到可表示群的简单嵌入。本文引入了群簇的动词等周函数的概念。证明了如果相对自由群的言语Dehn函数由多项式有界，则此群可拟等距嵌入到一个具有多项式等周函数的拟等距表示群中.利用多项式Dehn函数构造了任意Baumslag-Solitar可解群到一个可表示群的简单嵌入.
In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
Embedding finitely generated free-by-cyclic groups in {finitely generated free}-by-cyclic groups
本文改进了Feighn-Handel关于自由群映射筒子群的结果，证明有限生成自由-由循环群可嵌入到有限生成自由-由循环群中，并在超几何情况下作为准凸子群嵌入。
Spectra of finite symplectic and orthogonal groups
有限群的谱是它的元素阶的集合。得到了有限辛群和正交群的算术描述。特别地，得到了所有有限单单群和正交群的谱的描述。
Finitely generated groups with polynomial index growth
证明了一个可解的可解剩余有限群具有多项式指数增长的充要条件是它是极大极小群。我们还证明了，如果一个具有PIG的有限生成群是剩余有限可解的，那么它是一个线性群。这些结果特别适用于有界生成群，它们意味着每一个无限BG剩余有限群有一个无限线性商。
Finitely generated infinite torsion groups that are residually finite simple
本文证明每个有限生成剩余有限挠群可嵌入某个有限生成且剩余有限单的挠群，从而构造出有限生成无限挠的剩余有限单群，回答了 Olshanskii 与 Osin 的问题。
Realizing residually finite groups as subgroups of branch groups
本文证明任意有限生成剩余有限群均可嵌入保持挠性、可顺从性等性质的有限生成完全分支群，并据此构造有限生成不可顺从挠分支群。
Finitely generated (non-abelian) free metabelian pro-p groups, and wreath products of f.g. free abelian pro-p groups, are all finitely axiomatizable in the class of all profinite groups.
On finite groups with the Magnus Property
本文研究具有Magnus性质的有限群，证明了有限MP群可解，确定了有限本原MP群及其主因子的可能阶数，并刻画了有限本原群的MP直积与单块本原群的冠基幂。
The supremum of lengths of strict chains of nested centralizers is called the $c$-dimension (centralizer dimension) of $G$. We prove two structure theorems for locally finite groups of finite $c$-dimension. We also prove that the $c$-dimension of the quotient of such a group by a locally soluble radical is bounded in terms of the $c$-dimension of the group itself.
Deflating infinite Coxeter groups to finite groups
本文考虑了一个较小的版本的26个节点的介绍Bimonster组。
Finite simple groups of Lie type as expanders
有限李型单群作扩张子
Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group.
The full automorphism group of the power (di)graph of a finite group
我们刻画了有限群的幂（di）图的全自同构群。作为应用，我们解决了Doostabadi，Erfanian和Jafarzadeh在2013年提出的一个猜想。
Finite p-groups with small automorphism group
对于每个素数$p$，我们构造一个有限$p$ -群的族$\{G_i\}$，使得$|\Aut (G_i)|/|G_i|$变为$0$，因为$i$变为无穷大.这证明了一个著名的猜想，即$|G|$对每一个非交换的有限$p$ -群$G$整除$|\Aut(G)|$。
The Grushko decomposition of a finite graph of finite rank free groups: an algorithm
一个有限生成群允许一个分解，称为它的Grushko分解，成为自由不可分解群的一个自由积。给出了有限秩自由群的有限图的Grushko分解的一个算法。特别是，可以确定这样的组是否是自由的。
Virtual finite quotients of finitely generated groups
若G是N与H的半直积，且N是正规的，且N是N-生成的，则G具有如下性质：每个有限群是G的某个有限指数子群的商当且仅当N和H之一具有此性质.这也适用于3-流形和循环表示群，例如对于任意双曲3-流形M和任意有限单群S，存在M的循环覆盖，其基本群覆盖于S。当同调上的诱导映射是内射时，我们也给出了有限秩自由群的升HNN扩张的剩余有限性的一个简短证明。
On the finite images of finitely generated perfect groups
研究猜想存在一个有限生成完全群可将所有有限 d 生成完全群作为同态像，并对具有有界长度次正规级数、因子为阿贝尔群或半单群的情形给出证明。
Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, $\textrm{Out}(F_n)$, of the free group of rank $n$ is either virtually finitely generated abelian or contains a nonabelian free group. In this note we consider the more general situation of the outer automorphism group $\textrm{Out}(G)$ of a finitely generated free-by-finite group $G$. We show that $\textrm{Out}(G)$ is translation discrete and that every subgroup of $\textrm{Out}(G)$ is either virtually finitely generated abelian or contains a nonabelian free group.
We study the structure of discrete subgroups of the group $G[[r]]$ of complex formal power series under the operation of composition of series. In particular, we prove that every finitely generated fully residually free group is embeddable to $G[[r]]$.
A Reidemeister-Schreier theorem for finitely $L$-presented groups
我们证明了著名的Reidemeister-Schreier定理的一个变种，用于证明$L$ -呈现群。更精确地说，我们证明了一个有限的$L$ -呈现群的每个有限指数子群本身是有限的$L$ -呈现。我们的证明是建设性的，它产生一个有限的$L$ -介绍的子群。本文进一步研究了不变$L$ -表现群的有限指数子群是不变$L$ -表现群的条件。
Strongly indecomposable finite groups
受无限群理论中例子的启发，我们将子群不能分解的有限群分类为直积。
This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenabil…
In this paper some properties of the $\mathfrak{F}^*$-hypercenter of a finite group are studied where $\mathfrak{F}^*$ is the class of all finite quasi-$\mathfrak{F}$-groups for a hereditary saturated formation $\mathfrak{F}$ of finite groups. In particular, it is shown that the quasinilpotent hypercenter of a finite group coincides with the intersection of all maximal quasinilpotent subgroups.
Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A \ast B$. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to $A$ and $B$. This proves a conjecture of Tamburini and Wilson.
Virtually free pro-p groups whose torsion elements have finite centralizer
一个扭元具有有限个中心化子的拟自由pro-p群是有限个p-群与一个自由pro-p因子的自由pro-p积.
We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed g…
The Codegree Isomorphism Conjecture for Finite Simple Groups
该研究证明有限单群的 codegree 同构猜想，得出非阿贝尔有限单群可由其不可约特征标 codegree 集合在有限群中唯一确定的结论。
On the structure of finite groups isospectral to finite simple groups
有限群称为等谱群，如果它们具有相同的元素阶集。有限非交换单群$L$称为几乎可谱群，如果每一个与$L$等谱的有限群都是几乎单群，其基柱同构于$L$ .已知除$J_2$，$A_6$，$A_{10}$和$^3D_4(2)$外，所有有限的李型散生、交错和例外单群几乎都是谱可识别的.本文是证明V.D.马祖罗夫：存在一个正整数$d_0$，使得维数大于$d_0$的有限单经典群几乎可由谱识别.也就是说，我们证明了一个与维数至少为10的有限单辛群或正交群$L$等谱的~有限群的非交换合成因子要么同构于$L$，要么不是一个具有与$L$相同特征的Lie型群，并将这个结果与以前的工作相结合，我们推出Mazurov猜想在$d_0=60$中成立.
Endotrivial modules for finite groups schemes II
众所周知，如果G是有限群，则其核平凡模群是核平凡生成的.本文证明了对任意有限群概型G，对任意固定整数n > 0，n维模的核平凡模的同构类只有1000个.这为有限群概型的模群总是非平凡生成的猜想提供了证据。当G是一个与代数群相关联的无穷小群概型时，这个结果对于提升和扭曲endovrival模的结构问题也有一定的应用.
In the Kourovka notebook the following question is posed 14.101: Is it true that a periodic group saturated with finite simple groups of Lie type whose ranks are bounded in the aggregate, is itself a simple group of Lie type? In this paper we give a partial answer to this question for periodic groups saturated by finite simple groups of Lie type rank 1.
A finitely generated solvable group with unbounded iterated identity is constructed.
The aim of this paper is to compare and contrast the class of residually finite groups with the class of equationally Noetherian groups - groups over which every system of coefficient-free equations is equivalent to a finite subsystem. It is easy to construct groups that are residually finite but not equationally Noetherian (e.g. the direct sum of all finite groups) or vice versa (e.g. the additive group $(\mathbb{Q},+)$ of the rationals). However, no explicit examples that are finitely generated seem to appear in the literature. In this paper, we show that among finitely generated groups, the…
Let $λ(G)$ be the maximum number of subgroups in an irredundant covering of a finite group $G$. We prove that the finite groups with $λ(G)=|G|-t$, where $t\leq 5$, are solvable, and classify such groups.
A subgroup of a finite group is wide if each prime divisor of the group order divides the subgroup order. We obtain the description of finite soluble groups with no wide subgroups. We also prove that a finite soluble group with nilpotent wide subgroups has the quotient group by its hypercenter with no wide subgroups.
The Fesenko groups have finite width
证明了作为Nottingham群的自然闭子群出现的Fesenko群具有有限宽度和无限连续性。
Palindromic Width of Finitely Generated Solvable Groups
研究了可解群生成的可解群的回文宽度。我们证明了每一个可生成的$3$ -步可解群都有有限的回文宽度。更一般地，我们证明了有限的回文宽度的交换由幂零由幂零群。对任意可解群的步$\geq 3$，证明了如果$G$是一个可解群的一个可解生成群，且该可解群是一个阿贝尔群通过一个满足正规子群极大条件的群的扩张，则$G$的回文宽度是有限的.我们还证明了$\mathbb Z \wr \mathbb Z$关于标准生成元集的回文宽度是$3$。
In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds for arbitrary finite groups.
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