Stable commutator length in subgroups of PL(I)

具有换位子群有限性条件的群

Groups with Finiteness Conditions on commutator subgroups

本文研究了特定换位子群集合为有限的群的结构，重点分析了这类群与具有有限导出子群的群之间的关系。

On finite capable $p$-groups of class 2 with cyclic commutator subgroups

We study finite capable $p$-groups $G$ of nilpotency class 2 such that the commutator subgroup $γ_2(G)$ of $G$ is cyclic and the center of $G$ is contained in the Frattini subgroup of $G$.

Commutators of Sylow subgroups of alternating and symmetric groups, commutator width in the wreath product of groups

This paper investigate bounds of the commutator width \cite {Mur} of a wreath product of two groups. The commutator width of direct limit of wreath product of cyclic groups are found. For given a permutational wreath product sequence of cyclic groups we investigate its commutator width and some properties of its commutator subgroup. It was proven that the commutator width of an arbitrary element of the wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N} $ equals to 1. As a corollary, it is shown that the commutator width of Sylows $p$-subgroups of symmetric and alternating groups $p…

The commutator and centralizer description of Sylow 2-subgroups of alternating and symmetric groups

Given a permutational wreath product sequence of cyclic groups of prime order we research a commutator width of such groups and some properties of its commutator subgroup. Commutator width of Sylow 2-subgroups of alternating group $A_{2^{k}}$, permutation group $S_{2^k}$ and $C_p \wr B$ were founded. The result of research was extended on subgroups $(Syl_2 {A_{2^k}})'$, $p>2$. The paper presents a construction of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of $A_{2^k}$ were founded. Elements presentation of $(Syl_…

Commutator subgroup of the Hecke group $G_5$ is not congruence

We prove that the power subgroup $G_5^5$ and the commutator subgroup $G_5'$ of the Hecke group $G_5$ are not congruence.

A Sufficient Condition for Nilpotency of the Commutator Subgroup

Let $G$ be a finite group with the property that if $a,b$ are commutators of coprime orders, then $|ab|=|a||b|$. We show that $G'$ is nilpotent.

Local indicability and commutator subgroups of Artin groups

We determine which of the finite-type Artin groups are locally indicable, and compute presentations for their commutator subgroups.

On the commutator subgroup of a right-angled Artin group

We use polyhedral product models to analyse the structure of the commutator subgroup of a right-angled Artin group. In particular, we provide a minimal set of generators for the commutator subgroup, consisting of special iterated commutators of canonical generators.

Bounded conjugacy classes, commutators, and approximate subgroups

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup $G'$ is finite. We establish the following results. Let $K,n$ be positive integers and $G$ a group having a $K$-approximate subgroup $A$. If $|a^G|\leq n$ for each $a\in A$, then the commutator subgroup of $\langle A^G\rangle$ has finite $(K,n)$-bounded order. If $|[g,a]^G|\leq n$ for all $g\in G$ and $a\in A$, then the commutator subgroup of $[G,A]$ has fin…

Commutators in finite $p$-groups with $2$-generator derived subgroup

Let $G$ be a finite $p$-group whose derived subgroup $G'$ can be generated by $2$ elements. If $G'$ is abelian, Guralnick proved that every element of $G'$ is a commutator. In this paper, we prove that the condition that $G'$ should be abelian is not needed. Even more, we prove that every element of $G'$ is a commutator of the form $[x,g]$ for a fixed $x\in G$.

Permutation representations of the braid group commutator subgroup

We study the representations of the commutator subgroup of the braid group with n strands in the symmetric group of degree r. Motivated by some experimental results, we conjecture that for n>r, every such representation is trivial.

On finite groups in which commutators are covered by Engel subgroups

Let $m,n$ be positive integers and $w$ a multilinear commutator word. Assume that $G$ is a finite group having subgroups $G_1,\ldots,G_m$ whose union contains all $w$-values in $G$. Assume further that all elements of the subgroups $G_1,\ldots,G_m$ are $n$-Engel in $G$. It is shown that the verbal subgroup $w(G)$ is $s$-Engel for some $\{m,n,w\}$-bounded number $s$.

Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups

We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the commutator subgroup of a graph product group, and provide an explicit minimal set of generators for the commutator subgroup of a right-angled Coxeter group.

The commutator subgroups of free groups and surface groups

A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a surface group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.

On a reformulation of the commutator subgroup

For semigroup $S$, a commutative congruence $σ_{orient}$ on $S$ and a subsemigroup Orientable($S$) of $S$ were introduced in "Two cancellative commutative congruences and group diagrams", Semigroup Forum (2011) 82: 338-353. Here we demonstrate that when the semigroup is in fact a group $G$, then Orientable($G$) is the commutator subgroup $[G,G]$ and $ G / σ_{orient}$ is the abelian quotient group $G / [G,G]$.

On symmetric commutator subgroups, braids, links and homotopy groups

In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for braid and surface groups.

Commutator Subgroups of Welded Braid Groups

Let $WB_n$ be the welded (or loop) braid group on n strands, $n \geq 3$. We investigate commutator subgroup of $WB_n$. We prove that the commutator subgroup $WB_n'$ is finitely generated and Hopfian. We show that $WB_n'$ is perfect if and only if $n \geq 5$. We also compute finite presentation for $FWB_n'$, the commutator subgroup of the flat welded braid group $FWB_n$. Along the way, we investigate adorability of these groups.

Smallest nontrivial quotients of the commutator subgroup of braid groups

We prove that the smallest non-trivial quotients of the commutator subgroups of the braid groups are the alternating groups, proving a conjecture of Chudnovsky-Kordek-Li-Partin. Furthermore, we show that any minimal quotient map is the standard projection, composed with an automorphism of the alternating group.

A focal subgroup theorem for outer commutator words

Let $G$ be a finite group of order $p^am$, where $p$ is a prime and $m$ is not divisible by $p$, and let $P$ be a Sylow $p$-subgroup of $G$. If $w$ is an outer commutator word, we prove that $P\cap w(G)$ is generated by the intersection of $P$ with the set of $m$th powers of all values of $w$ in $G$

On finite groups in which coprime commutators are covered by few cyclic subgroups

The coprime commutators $γ_j^*$ and $δ_j^*$ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. They are defined as follows. Let $G$ be a finite group. Every element of $G$ is both a $γ_1^*$-commutator and a $δ_0^*$-commutator. Now let $j\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $γ_{j-1}^*$-commutators. An element $g$ is a $γ_j^*$-commutator if there exist $a\in X$ and $b\in G$ such that $g=[a,b]$ and $(|a|,|b|)=1$. For $j\geq 1$ let $Y$ be the set of all elements of …

Procyclic coverings of commutators in profinite groups

We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G' such that the order of M is m-bounded and G'/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G' is either finite of m-bounded order or procyclic.

辫子群交换子子群的自同构群

Automorphism group of the commutator subgroup of the braid group

本文研究辫子群交换子子群$B'_n$的自同构群，证明$n\ge4$时其自同构群等于$B_n$的自同构群，解决了Vladimir Lin的问题。

Conjugation-invariant norms on the commutator subgroup of the infinite braid group

In this paper, we give a proof of the result of Brandenbursky and Kȩdra which says that the commutator subgroup of the infinite braid group admits stably unbounded norms. Moreover, we observe the norms which we constructed are equivalent to the biinvariant word norm studied by Brandenbursky and Kȩdra.

Commutator Subgroups of Virtual and Welded Braid Groups

Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $VB_n'$ is finitely generated if and only if $n \geq 4$, and $WB_n'$ is finitely generated for $n \geq 3$. Also we prove that $VB_3'/VB_3'' =\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z}^{\infty}$, $VB_4' / VB_4'' = \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$, $WB…

The commutator subgroup of the braid group is generated by two elements

For $n$ at least 7 and $n$ equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on $n$ strands. These generating sets are of the smallest possible cardinality. For $n$ equal to 4 or 6, we give generating sets of size three. We also prove that the commutator subgroup of the braid group on 4 strands cannot be generated by fewer than three elements.

Large scale geometry of commutator subgroups

Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary word-hyperbolic group, C is one-ended. Hence (in this case), the asymptotic dimension of C is at least 2.

Characters, Commutators and Centers of Sylow Subgroups

The character table of a finite group G determines whether |P:P'|=p^2 and whether |P:Z(P)|=p^2, where P is a Sylow p-subgroup of G. To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.

由因子映射和子群胚产生的拓扑全群的最大子群

Maximal subgroups of topological full groups arising from factor maps and subgroupoids

本文通过因子映射和宽开子群胚构造拓扑全群及其换位子群的最大子群，给出最大性判据，并应用于康托尔极小系统，产生 Grigorchuk-Vorobets 猜想中的划分稳定化子及顺从群中的最大子群例子。

Commutators in finite p-groups with 3-generator derived subgroup

It is well known that, in general, the set of commutators of a group $G$ may not be a subgroup. Guralnick showed that if $G$ is a finite $p$-group with $p\ge 5$ such that $G'$ is abelian and $3$-generator, then all the elements of the derived subgroup are commutators. In this paper, we extend Guralnick's result by showing that the condition of $G'$ to be abelian is not needed. In this way, we complete the study of this property in finite $p$-groups in terms of the number of generators of the derived subgroup. We will also see that the same result is true when the action of $G$ on $G'$ is unise…

Homomorphisms of commutator subgroups of braid groups

We give a complete classification of homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $n$ strands when $n$ is at least 7. In particular, we show that each nontrivial homomorphism extends to an automorphism of the braid group on $n$ strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.

Inclusion among commutators of elementary subgroups

In the present paper we continue the study of the elementary commutator subgroups $[E(n,A),E(n,B)]$, where $A$ and $B$ are two-sided ideals of an associative ring $R$, $n\ge 3$. First, we refine and expand a number of the auxiliary results, both classical ones, due to Bass, Stein, Mason, Stothers, Tits, Vaserstein, van der Kallen, Stepanov, as also some of the intermediate results in our joint works with Hazrat, and our own recent papers [40,41]. The gimmick of the present paper is an explicit triple congruence for elementary commutators $[t_{ij}(ab),t_{ji}(c)]$, where $a,b,c$ belong to three …

On nilpotency of higher commutator subgroups of a finite soluble group

Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $δ_k$-values $a,b\in G$ of coprime orders. In the course of the proof we establish the following result of independent interest: Let $P$ be a Sylow $p$-subgroup of $G$. Then $P\cap G^{(k)}$ is generated by $δ_k$-values contained in $P$. This is related to the so-called Focal Subgroup Theorem.

Special subgroups of gyrogroups: Commutators, nuclei and radical

A gyrogroup is a nonassociative group-like structure modelled on the space of relativistically admissible velocities with a binary operation given by Einstein's velocity addition law. In this article, we present a few of groups sitting inside a gyrogroup $G$, including the commutator subgyrogroup, the left nucleus, and the radical of $G$. The normal closure of the commutator subgyrogroup, the left nucleus, and the radical of $G$ are in particular normal subgroups of $G$. We then give a criterion to determine when a subgyrogroup $H$ of a finite gyrogroup $G$, where the index $[G\colon H]$ is th…

Central quotient versus commutator subgroup of groups

In 1904, Issai Schur proved the following result. If $G$ is an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$, then the commutator subgroup of $G$ is finite. A partial converse of this result was proved by B. H. Neumann in 1951. He proved that if $G$ is a finitely generated group with finite commutator subgroup, then $G/\Z(G)$ is finite. In this short note, we exhibit few arguments of Neumann, which provide further generalizations of converse of the above mentioned result of Schur. We classify all finite groups $G$ such that $|G/\Z(G)| = |γ_2(G…

有限正交群换位子群中的双反射性

Bireflectionality in the commutator subgroup of a finite orthogonal group

该研究针对特征非2的有限域上正交群换位子群，分类了其中的双反射与可逆元素，明确了可逆元为双反射的判定条件及例外情形。

On profinite groups with commutators covered by countably many cosets

Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the following theorem. Let C be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal C-subgroups is again a C-subgroup. If w is a multilinear commutator and G is a profinite group such that the set of all w-values is contained in a union o…

实数域上正交群换位子群的双反射

Bireflections of the commutator subgroup of an orthogonal group over the reals

本文研究实数域上符号为$(p,q)$的正交群$O(p,q)$的换位子群$O(p,q)'$，证明其中元素为双反射元当且仅当可逆，并对双反射元素进行了分类。

Polyhedral products, and relations in the commutator subgroup of a right-angled Coxeter group

We give a criterion of the existence of a presentation with a single relation for the commutator subgroup $RC_{\mathscr{K}}'$ of a right-angled Coxeter group $RC_{\mathscr{K}}$. Namely, we prove that $RC_{\mathscr{K}}'$ is a one-relator group if and only if $\mathscr{K}$ is either a $p$-cycle with $p\geqslant4$ or has the form ($p$-cycle)$*Δ^q$ with $p\geqslant4$ and $g\geqslant0$, where $Δ^q$ is a $q$-simplex.

Coverings of commutators in profinite groups

Let $w$ be a group-word. Suppose that the set of all $w$-values in a profinite group $G$ is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup $w(G)$ depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to …

Stable commutator length on big mapping class groups

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.

Commutator subgroups of Sylow 2-subgroups of alternating group and Miller-Moreno groups as bases of new Key Exchange Protocol

The goal of this investigation is effective method of key exchange which based on non-commutative group $G$. The results of Ko et al. \cite{kolee} is improved and generalized. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group is found. The structure of the commutator subgroup of Sylow 2-subgroups of the alternating group ${A_{2^{k}}}$ is investigated and used in key exchange protocol which based on non-commutative group. We consider non-commutative generalization of CDH problem \cite{gu2013new, bohli2006towards} on base of metacyclic gro…

Algebraic commutators with respect to subnormal subgroups in division rings

Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily contained in the center $F$ of $D$. In this paper, we study the structure of $D$ under the condition of left algebraicity of certain subsets of $D$ over $K$. Among results, it is proved that if $D^*$ contains a noncentral normal subgroup which is left algebraic over $K$ of bounded degree $d$, then $[D:F]\le d^2$. In case $K=F$, the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of $D^*$ are algebraic of bounded degree $d$…

BFC-theorems for higher commutator subgroups

A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954 B. H. Neumann discovered that if G is a BFC-group then the derived group G' is finite. Let w=w(x_1,\dots,x_n) be a multilinear commutator. We study groups in which the conjugacy classes containing w-values are finite of bounded order. Let G be a group and let w(G) be the verbal subgroup of G generated by all w-values. We prove that if x^G has size at most m for every w-value x, then the derived subgroup of w(G) is finite of order bounded by a function of m and n. If x^{w(G)} has size at most m for every…

On profinite groups in which commutators are covered by finitely many subgroups

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contain…

Words and pronilpotent subgroups in profinite groups

Let $w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

受限子群的交换子引理及其在作用于根树上的群中的应用

A commutator lemma for confined subgroups and applications to groups acting on rooted trees

本文证明受限子群的交换子引理，并据此刻画弱分支群的 URS 结构，进而证明有限生成分支群作用增长最小、嵌入强有界型同胚群必空间实现以及不嵌入 IET 群等结论。

Commutator Subgroups of Twin Groups and Grothendieck's Cartographical Groups

Let $TW_n$ be the twin group on $n$ arcs, $n \geq 2$. The group $TW_{m+2}$ is isomorphic to Grothendieck's $m$-dimensional cartographical group $\mathcal C_m$, $m \geq 1$. In this paper we give a finite presentation for the commutator subgroup $TW_{m+2}'$, and prove that $TW_{m+2}'$ has rank $2m-1$. We derive that $TW_{m+2}'$ is free if and only if $m \leq 3$. From this it follows that $TW_{m+2}$ is word-hyperbolic and does not contain a surface group if and only if $m \leq 3$. It also follows that the automorphism group of $TW_{m+2}$ is finitely presented for $m \leq 3$.

Structure and minimal generating sets of Sylow 2-subgroups of alternating groups, properties of its commutator subgroup

In this article the investigation of Sylows p-subgroups of ${{A}_{n}}$ and ${{S}_{n}}$, which was started in article of U. Dmitruk, V. Suschansky "Structure of 2-sylow subgroup of symmetric and alternating group" and article of R.~Skuratovskii "Corepresentation of a Sylow p-subgroup of a group S_n" \cite{Dm, Sk, Paw} is continued. Let $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ be Sylow 2-subgroups of corresponding alternating groups $A_{2^k}$ and $A_{n}$. We find a least generating set and a structure for such subgroups $Sy{{l}_{2}}{{A}_{{{2}^{k}}}}$ and $Syl_2{A_{n}}$ and commutator width of $Syl_2{…

关于正规子群的简洁性及由外交换子词产生的新简洁词

Conciseness on normal subgroups and new concise words from outer commutator words

本文通过研究外交换子词在正规子群上的取值，证明了当各变量集互不相交的非交换子词代入外交换子词时所得词是简洁的，并推广到幂次形式，同时建立了正规子群上取值集有限则生成子群有限的结果。

Commutators in groups of piecewise projective homeomorphisms

In 2012 Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and later Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups $H$ and $G_0$. We show that the group $H$ of piecewise projective homeomorphisms of $\mathbb{R}$ has the property that $H"$ is simple and that every proper quotient of $H$ is metabelian. We establish simplicity of the commutator s…

Commutator width in the first Grigorchuk group

Let $G$ be the first Grigorchuk group. We show that the commutator width of $G$ is $2$: every element $g\in [G,G]$ is a product of two commutators, and also of six conjugates of $a$. Furthermore, we show that every finitely generated subgroup $H\leq G$ has finite commutator width, which however can be arbitrarily large, and that $G$ contains a subgroup of infinite commutator width. The proofs were assisted by the computer algebra system GAP.

Commutators in groups definable in o-minimal structures

We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.

Generation of relative commutator subgroups in Chevalley groups. II

In the present paper, which is a direct sequel of our paper [12] joint with Roozbeh Hazrat, we prove unrelativised version of the standard commutator formula in the setting of Chevalley groups. Namely, let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(Φ,R)$ of type $Φ$ over $R$. The unrelativised elementary subgroup $E(Φ,I)$ of level $I$ is generated (as a group) by the elementary unipotents $x_α(ξ)$, $α\inΦ$, $ξ\in I$, of level $I$. Obviously, in general $E(Φ,I)$ has n…

对称群虚拟扩张的换位子群与晶体学商群

Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups

本文研究对称群虚拟扩张的换位子群与晶体学商群，给出纯虚拟三重群及两个换位子群的显式有限表示，并证明 $VL_n/PVL_n^{'}$ 为晶体学群且构造了无穷多个挠元素。