33 pages, no figure, Proposition 4.4 in the previous version was false, and we replaced it with a weaker version. To appear in Kodai Mathematical Journal
In the present paper, for a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the mixed commutator length $\mathrm{cl}_{G,N}$ on the mixed commutator subgroup $[G,N]$. We focus on the setting of wreath products: $ (G,N)=(\mathbb{Z}\wr Γ, \bigoplus_Γ\mathbb{Z})$. Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group $Γ$ is not locally cyclic, the ordinary commutator length $\mathrm{cl}_G$ does not coincide with $\mathrm{cl}_{G,N}$ on $[G,N]$ for the above pair. On the other hand, we prove that if $Γ$ is locally cyclic, then for every pair $(G,N)$ such that $1\to N\to G\to Γ\to 1$ is exact, $\mathrm{cl}_{G}$ and $\mathrm{cl}_{G,N}$ coincide on $[G,N]$. We also study the case of permutational wreath products when the group $Γ$ belongs to a certain class related to surface 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 size of minimal generating set for commutator of Sylow 2-subgroup of alternating group was found. Given a permutational wreath product of finite cyclic groups sequence we prove that the commutator width of such groups is 1 and we research some properties of its commutator subgroup. It was shown that $(Syl_2 A_{2^k})^2 = Syl'_2 (A_{2^k}), \, k>2$. A new approach to presentation of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$ was applied. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$, permutation group ${S_{2^{k}}}$ and …
Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author \cite{SkVC, SkMal, SkAr} and construct minimal generating set for wreath product of finite and infinite cyclic groups and direct product of such groups. We generalize results of Meldrum about commutator subgroup of wreath product \cite{Meld} because we take in consideration as regular wreath product as well as no regular (where active group $\mathcal{A}$ can acts not…
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_…
We prove that the nilpotent product of a set of groups $A_{1},\dots, A_{s}$ has finite palindromic width if and only if the palindromic widths of $A_{i}, i=1,\dots, s,$ are finite. We give a new proof that the commutator width of $F_n \wr K$ is infinite, where $F_n$ is a free group of rank $n \geq 2$ and $K$ a finite group. This result, combining with a result of Fink \cite{f1} gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.
We study the unit group of the modular group algebra KG, where G is a 2-group of maximal class. We prove that the unit group of KG possesses a section isomorphic to the wreath product of a group of order two with the commutator subgroup of the group G.
We show that the wreath product $G \wr \mathbb{Z}^n$ of any finitely generated group $G$ with $\mathbb{Z}^n$ has finite palindromic width. We also show that $C \wr A$ has finite palindromic width if $C$ has finite commutator width and $A$ is a finitely generated infinite abelian group. Further we prove that if $H$ is a non-abelian group with finite palindromic width and $G$ any finitely generated group, then every element of the subgroup $G' \wr H$ can be expressed as a product of uniformly boundedly many palindromes. From this we obtain that $P \wr H$ has finite palindromic width if $P$ is a …
Let $G_{p,d}$ be the restricted wreath product $C_{p} wr C^{d}$ where $C_{p}$ is a cyclic group of order a prime $p$ and $C^{d}$ a free abelian group of finite rank $d$. We study the existence of faithful state-closed (fsc) representations of $G_{p,d}$ on the $1$-rooted $m$-ary tree for some finite $m$. The group $G_{2,1}$, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for $d \geq 2$ there are no fsc representations of $G_{p,d}$ on the $p$ -adic tree. We characterize all fsc representations of $G=G_{p,1}$ on the $p$-adic tree where the first lev…
Let $\mathcal{S}$ be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in $\mathcal{S}$ is topologically finitely generated, provided that the actions of the groups in $\mathcal{S}$ are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in $\mathcal{S}$ are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterat…
Let $N$ be a normal subgroup of a group $G$. A quasimorphism $f$ on $N$ is $G$-invariant if $f(gxg^{-1}) = f(x)$ for every $g \in G$ and every $x \in N$. The goal in this paper is to establish Bavard's duality theorem of $G$-invariant quasimorphisms, which was previously proved by Kawasaki and Kimura in the case $N = [G,N]$. Our duality theorem provides a connection between $G$-invariant quasimorphisms and $(G,N)$-commutator lengths. Here for $x \in [G,N]$, the $(G,N)$-commutator length $\mathrm{cl}_{G,N}(x)$ of $x$ is the minimum number $n$ such that $x$ is a product of $n$ commutators which …
It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k, y_k] = z^n$, where $n \ge 2k$, in a free product $\mathcal{F}$ of groups without nontrivial elements of order $\le n$ implies that $z$ is conjugate to an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct.
We investigate closure results for C-approximable groups, for certain classes C of groups with invariant length functions. In particular we prove, each time for certain (but not necessarily the same) classes C that (i) the direct product of two C-approximable groups is C-approximable; (ii) the restricted standard wreath product G wr H is C-approximable when G is C-approximable and H is residually finite; and (iii) a group G$ with normal subgroup N is C-approximable when N is C-approximable and G/N is amenable. Our direct product result is valid for LEF, weakly sofic and hyperlinear groups, as …
In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. We prove that for any group $G\in\mathcal{G}$ the ranks of the center $Z(G)$ and the quotient by commutator subgroup $G/[G,G]$ coincide. Moreover, this rank is a first Betti number of the orbit of Morse function.
Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group P and stabilizing gates within an extension group C. In this paper one explores the nice adequacy between group theoretical concepts such as commutators, normal subgroups, group of automorphisms, short exact sequences, wreath products... and the coherent quantum computational primitives. The structure of the single qubit and two-qubit Clifford groups is analyzed in detai…
We construct invariant quasimorphisms for groups acting on the circle. Furthermore, we provide a criterion for the non-extendablity of the resulting quasimorphisms and an explicit formula which relates the values of our quasimorphisms to those of the Poincaré translation number. By using them, we show that the stable commutator length $\mathrm{scl}_G$ and the stable mixed commutator length $\mathrm{scl}_{G,N}$ are not bi-Lipschitzly equivalent for the surface group $G=π_1(Σ_{\ell})$ of genus at least $2$ and its commutator subgroup $N = [π_1(Σ_{\ell}), π_1(Σ_{\ell})]$. We also show the non-equ…
In the paper we discuss the algebraic structure of topological full group $[[T]]$ of a Cantor minimal system $(X,T)$. We show that the topological full group $[[T]]$ has the structure similar to a union of permutational wreath products of group $\mathbb Z$. This allows us to prove that the topological full groups are locally embeddable into finite groups; give an elmentary proof of the fact that group $[[T]]'$ is infinitely presented; and provide explicit examples of maximal locally finite subgroups of $[[T]]$. We also show that the commutator subgroup $[[T]]'$, which is simple and finitely-ge…
