Alireza Abdollahi
In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime $p$ and any left 3-Engel element $x$ of finite $p$-power order in a group $G$, $x^p$ is in the Baer radical of $G$. Also it is proved that $<x,y>$ is nilpotent of class 4 for every two left 3-Engel elements in a group $G$.
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Left $3$-Engel elements in groups: A survey
该论文为一篇综述类研究，聚焦群论领域中的左3-Engel元相关内容，对该方向的研究情况进行了梳理总结。
We give an example of a locally nilpotent group $G$ containing a left $3$-Engel element $x$ where $\langle x\rangle^{G}$ is not nilpotent.
For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where $\langle x \rangle^G$ is not nilpotent.
Let $G$ be a group and let $x\in G$ be a left $3$-Engel element of order dividing $60$. Suppose furthermore that $\langle x\rangle^{G}$ has no elements of order $8$, $9$ and $25$. We show that $x$ is then contained in the locally nilpotent radical of $G$. In particular all the left $3$-Engel elements of a group of exponent $60$ are contained in the locally nilpotent radical.
In this paper we prove a group theoretic analogue of the well known local nilpotence theorem for sandwich Lie algebras due to Kostrikin and Zel'manov. We introduce the notion of a strong left 3-Engel element of a group G and show that these are always in the locally nilpotent radical of G. This generalises a previous result of Jabara and Traustason that showed that a left 3-Engel element a of a group G is in the locally nilpotent radical of G whenever a is of odd order.
Let $Γ$ be the first Grigorchuk group. According to a result of Bartholdi, the only left Engel elements of $Γ$ are the involutions. This implies that the set of left Engel elements of $Γ$ is not a subgroup. Of particular interest is to wonder whether this happens also for the sets of bounded left Engel elements, right Engel elements, and bounded right Engel elements of $Γ$. Motivated by this, we prove that these three subsets of $Γ$ coincide with the identity subgroup.
In this paper we study left and right 4-Engel elements of a group. In particular, we prove that $<a, a^b>$ is nilpotent of class at most 4, whenever $a$ is any element and $b^{\pm 1}$ are right 4-Engel elements or $a^{\pm 1}$ are left 4-Engel elements and $b$ is an arbitrary element of $G$. Furthermore we prove that for any prime $p$ and any element $a$ of finite $p$-power order in a group $G$ such that $a^{\pm 1}\in L_4(G)$, $a^4$, if $p=2$, and $a^p$, if $p$ is an odd prime number, is in the Baer radical of $G$.
We give an infinite family of examples that generalise the construction given in arXiv:1811.12074 of a locally finite 2-group $G$ containing a left 3-Engel element $x$ where ${\langle x \rangle}^G$, the normal closure of $x$ in $G$, is not nilpotent. The construction is based on a family of Lie algebras that are of interest in their own right and make use of a classical theorem of Lucas, regarding when $\binom{m}{n}$ is even.
The following problem is proposed as Problem 18.57 in [The Kourovka Notebook, No. 18, 2014] by D. V. Lytkina: Let $G$ be a finite $2$-group generated by involutions in which $[x, u, u] = 1$ for every $x \in G$ and every involution $u \in G$. Is the derived length of $G$ bounded? The question is asked of an upper bound on the solvability length of finite $2$-groups generated by involutions in which every involution (not only the generators) is also left $2$-Engel. We negatively answer the question.
We generalize to FC*, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002), 241-254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an FC*-group G coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of G are subgroups and the former coincides with the hypercentre. We also give an example of an FC*-group for which the set of right Engel elements con…
A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) We prove that if a profinite group $G$ admits a coprime automorphism $φ$ of prime order such that every fixed point of $φ$ has a finite right Engel sink, then $G$ has an open locally nilpotent subgroup. A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ a…
Suppose that a finite group $G$ admits a soluble group of coprime automorphisms $A$. We prove that if, for some positive integer $m$, every element of the centralizer $C_G(A )$ has a left Engel sink of cardinality at most $m$ (or a right Engel sink of cardinality at most $m$), then $G$ has a subgroup of $(|A|,m)$-bounded index which has Fitting height at most $2α(A)+2$, where $α(A)$ is the composition length of $A$. We also prove that if, for some positive integer $r$, every element of the centralizer $C_G(A )$ has a left Engel sink of rank at most $r$ (or a right Engel sink of rank at most $r…
We give a survey of results on the structure of right and left Engel elements of a group. We also present some new results in this topic.
We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all $p$-elements and that the group is virtually a $p$-group (and so periodic) for some prime $p$. We also show that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition. Also, we apply these resu…
We prove that the set of right 4-Engel elements of a group $G$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $<x>^G$ is nilpotent of class at most 7 for each right 4-Engel elements $x$ of $G$.
A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is a left Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) We prove that if a finite group $G$ admits an automorphism $φ$ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(φ)$ has a left Engel sink of cardinality at most $m$, then the index of the second Fitting subgroup $F_2(G)$ is bounded in terms of $m$. A r…
Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $r\geq2$ acting on a finite $q'$-group $G$. The following results are proved. We show that if all elements in $γ_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $a\in A^\#$, then $γ_{r-1}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $a\in A^\#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Assuming $r\geq 3$…
Let $R_n(G)$ denotes the set of all right $n$-Engel elements of a group $G$. We show that in any group $G$ whose 5th term of lower central series has no element of order 2, $R_3(G)$ is a subgroup. Furthermore we prove that $R_4(G)$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $<x >^G$ is nilpotent of class at most 7 for each $x\in R_4(G)$. Using a group constructed by Newman and Nickel we also show that, for each $n\geq 5$, there exists a nilpotent group of class $n+2$ containing a right $n$-Engel element $x$ and an el…
On finite groups containing an element whose Engel sink is small
本文利用有限简单群分类证明，若有限群G中存在元素g使得G=[G,g]，则G的阶数受g的右Engel sinks和左Engel sinks限制。
We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a prime and $A$ an elementary abelian $q$-group of order at least $q^2$ acting coprimely on a profinite group $G$. Assume that all elements in $C_{G}(a)$ are Engel in $G$ for each $a\in A^{\#}$. Then $G$ is locally nilpotent (Theorem B2). Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^3$ acting coprimely on a finite gr…
Let $q$ be a prime. Let $G$ be a residually finite group satisfying an identity. Suppose that for every $x \in G$ there exists a $q$-power $m=m(x)$ such that the element $x^m$ is a bounded Engel element. We prove that $G$ is locally virtually nilpotent. Further, let $d,n$ be positive integers and $w$ a non-commutator word. Assume that $G$ is a $d$-generator residually finite group in which all $w$-values are $n$-Engel. We show that the verbal subgroup $w(G)$ has $\{d,n,w\}$-bounded nilpotency class.
For an element $g$ of a group $G$, an Engel sink is a subset ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. A~finite group is nilpotent if and only if every element has a trivial Engel sink. We prove that if in a finite group $G$ every element has an Engel sink generating a subgroup of rank~$r$, then $G$ has a normal subgroup $N$ of rank bounded in terms of $r$ such that $G/N$ is nilpotent.
We say that an element $g$ of a group $G$ is almost right Engel if there is a finite set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$, that is, for every $x\in G$ there is a positive integer $n(x,g)$ such that $[...[[g,x],x],\dots ,x]\in {\mathscr R}(g)$ if $x$ is repeated at least $n(x,g)$ times. Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$. We prove that if all elements of a compact (Hausdorff) group $G$ are almost right Engel, then $G$ has a finite normal sub…
An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group $G$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. This settles a question suggested by J. S. Wilson.
Compact groups in which all elements have countable right Engel sinks
本文证明了若紧致群的每个元素具有可数右Engel汇，则该群存在有限正规子群使其商群局部幂零。
An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) It is proved that if a profinite group $G$ admits an elementary abelian group of automorphisms $A$ of coprime order $q^2$ for a prime $q$ such that for each $a\in A\setminus\{1\}$ every element of the centralizer $C_G(a)$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ i…
Let G be a linear group such that for every g in G there is a finite set R(g) with the property that for every x in G all sufficiently long commutators [g,x,x,...,x] belong to R(g). It is proved that G is finite-by-hypercentral.
An element $g$ of a group $G$ is said to be right Engel if for every $x\in G$ there is a number $n=n(g,x)$ such that $[g,{}_{n}x]=1$. We prove that if a profinite group $G$ admits a coprime automorphism $φ$ of prime order such that every fixed point of $φ$ is a right Engel element, then $G$ is locally nilpotent.
Finite and profinite groups with small Engel sinks of p-elements
本文证明有限群和profinite群中p-元素具有有界或有限Engel汇时，存在正规子群使得商群为p'-群，且指数或结构受控。
The main result of the paper is the following theorem. Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^2$. Suppose that $A$ acts coprimely on a finite group $G$ and assume that for each $a\in A^{\#}$ every element of $C_{G}(a)$ is $n$-Engel in $G$. Then the group $G$ is $k$-Engel for some $\{n,q\}$-bounded number $k$.
Let $p$ be a prime and let $G$ be a subgroup of a Sylow pro-$p$ subgroup of the group of automorphisms of the $p$-adic tree. We prove that if $G$ is fractal and $|G':\mathrm{st}_G(1)'|=\infty$, then the set $L(G)$ of left Engel elements of $G$ is trivial. This result applies to fractal nonabelian groups with torsion-free abelianization, for example the Basilica group, the Brunner-Sidki-Vieira group, and also to the GGS-group with constant defining vector. We further provide two examples showing that neither of the requirements $|G':\mathrm{st}_G(1)'|=\infty$ and being fractal can be dropped.
For an element $g$ of a group $G$, an Engel sink is a subset $\mathscr{E}(g)$ such that for every $ x\in G $ all sufficiently long commutators $ [x,g,g,\ldots,g] $ belong to $\mathscr{E}(g)$. Let $q$ be a prime, let $m$ be a positive integer and $A$ an elementary abelian group of order $q^2$ acting coprimely on a finite group $G$. We show that if for each nontrivial element $a$ in $ A$ and every element $g\in C_{G}(a)$ the cardinality of the smallest Engel sink $\mathscr{E}(g)$ is at most $m$, then the order of $γ_\infty(G)$ is bounded in terms of $m$ only. Moreover we prove that if for each $…
Centralizer-like Subgroups Associated with the $n$-Engel Words Inside of Direct Product Groups
本文研究直积群中与 $n$-Engel 字相关的类中心化子子群，证明两个群的直积的右 $n$-Engel 元素集合等于各直积因子的右 $n$-Engel 元素集合的直积，并提出后续研究问题。
Engel groups and Engel elements became popular in 50s. We consider in the paper the more general nil-groups and nil-elements in groups. All these notions are related to nilpotent groups and nilpotent radicals in groups. These notions generate problems which are parallel to Burnside problems for periodic groups. The first three theorems of the paper are devoted to nil-groups and Engel groups, while the other results are connected with the further generalizations. These generalizations extend the theory to solvable groups and solvable radicals in groups. The paper has two parts. The first one (s…
Let $q$ be a prime and $A$ an elementary abelian $q$-group acting as a coprime group of automorphisms on a profinite group $G$. We show that if $A$ is of order $q^2$ and some power of each element in $C_G(a)$ is Engel in $G$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Assuming that $A$ is of order $q^3$ we prove that if some power of each element in $C_G(a)$ is Engel in $C_G(a)$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Some analogues consequences of quantitative nature for finite groups are also obtained.
For an element $g$ of a group $G$, an Engel sink is a subset $\mathcal{E}(g)$ such that for every $ x\in G $ all sufficiently long commutators $ [x,g,g,\ldots,g] $ belong to $\mathcal{E}(g)$. We conjecture that if $G$ is a profinite group in which every element admits a sink that is a procyclic subgroup, then $G$ is procyclic-by-(locally nilpotent). We prove the conjecture in two cases -- when $G$ is a finite group, or a soluble pro-$p$ group.
We give an affrmative answer to the question whether a residually finite Engel group satisfying an identity is locally nilpotent. More generally, for a residually finite group G with an identity, we prove that the set of right Engel elements of G is contained in the Hirsch-Plotkin radical of G. Given an arbitrary word w, we also show that the class of all groups G in which the w-values are right n-Engel and w(G) is locally nilpotent is a variety.
For every Turing machine, we construct an automaton group that simulates it. Precisely, starting from an initial configuration of the Turing machine, we explicitly construct an element of the group such that the Turing machine stops if, and only if, this element is of finite order.If the Turing machine is universal, the corresponding automaton group has an undecidable order problem. This solves a problem raised by Grigorchuk.The above group also has an undecidable Engel problem: there is no algorithm that, given g, h in the group, decides whether there exists an integer n such that the n-itera…
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…
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 $s$.
Let $G$ be a profinite group in which for every element $x\in G$ there exists a natural number $q=q(x)$ such that $x^q$ is Engel. We show that $G$ is locally virtually nilpotent. Further, let $p$ be a prime and $G$ a finitely generated profinite group in which for every $γ_k$-value $x\in G$ there exists a natural $p$-power $q=q(x)$ such that $x^q$ is Engel. We show that $γ_k(G)$ is locally virtually nilpotent.
We consider decidability problems associated with Engel's identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given $x,y$, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, namely elements $y$ such that the map $x\mapsto[x,y]$ attracts to $\{1\}$. Although this problem seems intractable in general, we prove that it is decidable for Grigorchuk's group: Engel eleme…
For a finite group $G,$ we investigate the direct graph $Γ(G),$ whose vertices are the non-hypercentral elements of $G$ and where there is an edge $x\mapsto y$ if and only if $[x,_ny]=1$ for some $n \in \mathbb N.$ We prove that $Γ(G)$ is always weakly connected and is strongly connected if $G/Z_{\infty}(G)$ is neither Frobenius nor almost simple.
Let $m,n$ be positive integers. Suppose that $G$ is a residually finite group in which for every element $x \in G$ there exists a positive integer $q=q(x) \leqslant m$ such that $x^q$ is $n$-Engel. We show that $G$ is locally virtually nilpotent. Further, let $w$ be a multilinear commutator and $G$ a residually finite group in which for every product of at most $896$ $w$-values $x$ there exists a positive integer $q=q(x)$ dividing $m$ such that $x^q$ is $n$-Engel. Then $w(G)$ is locally virtually nilpotent.
On finitely generated Engel branch groups
本文构造了有限生成的Engel分支群，回答了相关存在性问题，并提供了继Golod之后第二类有限生成非幂零Engel群的例子，方法基于根树作用与词长快速收缩。
Let $G$ be a non-Engel group and let $L(G)$ be the set of all left Engel elements of $G$. Associate with $G$ a graph $\mathcal{E}_G$ as follows: Take $G\backslash L(G)$ as vertices of $\mathcal{E}_G$ and join two distinct vertices $x$ and $y$ whenever $[x,_k y]\not=1$ and $[y,_k x]\not=1$ for all positive integers $k$. We call $\mathcal{E}_G$, the Engel graph of $G$. In this paper we study the graph theoretical properties of $\mathcal{E}_G$.
Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,{}_{n}x]$ over $x\in G$. In the case when $G$ is soluble we prove that if, for some $n$, the Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$, it is proved that if, for some $n$, the generalized Fitting height of $R_n(g)$ is equal to $k$, then $g$ belongs to the generalized Fitting subgroup $F^*_{f(k,m)}(G)$ with $f(k,m)$ depending only on $k$ and $m$, where $|g|$ is the product of $m$ primes …
Compact groups in which commutators have finite right Engel sinks
该群论研究证明，满足k重换位子均有有限右Engel沉的紧群存在局部幂零开子群，且当右Engel沉基数不超过m时，该子群的有限指数仅由m界定。
A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimensional Lie algebra defined over a field of characteristic zero in similar terms. We suggest a conjectural description of the solvable radical of a finite group as the set of Engel-like elements and reduce this conjecture to the case of a finite simple group.
Given a finite group $G$, the Engel graph of $G$ is a directed graph encoding pairs of elements satisfying some Engel word. From the work of Detomi, Lucchini and Nemmi, the strongly connectivity of the Engel graph of an arbitrary group $G$ is reduced to the understanding of the strongly connectivity of the Engel graph of non-abelian simple groups. In this paper, we investigate the strongly connectivity of the Engel graph of finite non-abelian simple groups.
Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\leq m$ for all $g\in G$,…
On the strong connectivity of the 2-Engel graphs of almost simple groups
本文针对几乎单群的2-Engel图，细化Lucchini等人关于其强连通性的已有分析结果。
Let x be an element of a group G. For a positive integer n let E_n(x) be the subgroup generated by all commutators [...[[y,x],x],...,x] over y in G, where x is repeated n times. There are several recent results showing that certain properties of groups with small subgroups E_n(x) are close to those of Engel groups. The present article deals with orderable groups in which, for some n, the subgroups E_n(x) are polycyclic. Let h,n be positive integers and G an orderable group in which E_n(x) is polycyclic with Hirsch length at most h for every x in G. It is proved that there are (h,n)-bounded num…
On the diameter of Engel graphs
本文研究有限群 Engel 有向图在强连通情形下的直径，并给出一个不依赖具体群的绝对上界。
5-Engel Lie algebras II
本文研究了5-Engel Lie algebra中由元素生成的理想，证明了特征$p>3$时该理想幂零并给出最佳幂零类界限，指出特征2和3时不一定幂零。
We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of $X^*$. We describe algorithms answering the word problem, and bound its complexity under some additional assumptions. We give a partial algorithm that decides in a group generated by an automaton, given $x,y$, whether an Engel identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) is satisfied. This algorithm succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, nam…
Following Plotkin we say that the automorphism $x$ of the group $G$ is a nil-automorphism if, for every $g\in G$, there exists $n=n(g)$ such that $[g,_n x]=1$. If the integer $n$ can be chosen independently of $g$, then $x$ is said to be unipotent. Nil and unipotent automorphisms can be regarded as a natural extension of the concept of Engel element, since a nil-automorphism $x$ is just a left-Engel element in $G < x >$. In this paper we consider nil-automorphisms of groups with residual properties namely locally-graded groups, residually-finite groups and profinite groups. The first result we…
Surjectivity of Engel Words on $\mathrm{SL}_2(\mathcal{O})$ and $\mathrm{PSL}_2(\mathcal{O}_2)$
研究局部环上SL_2和PSL_2群的Engel词映射，证明在剩余域特征≠2且q足够大时，SL_2(k)的非标量元提升在SL_2(O)的m次Engel词像中，且所有Engel词映射在PSL_2(O_2)上满射，推广了域上的结果。
In part I it was shown that for each k>0 the generalized Sato-Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient G_k of its fundamental group, `functorially' invariant under k-quasi-isotopy. Here we show that Cochran's derived invariant β^k, provided k>2, and a series of \barμ-invariants, starting with \barμ(111112122) for k=3, also fall in this gap. In fact, all \barμ-invariants where each index occurs at most k+1 times, except perhaps for one occuring k+2 times, can be extracted from G_k, and if they vanish, G…
Let $g$ be an element of a finite group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots,g]$ over $x\in G$, where $g$ is repeated $n$ times. By Baer's theorem, if $E_n(g)=1$, then $g$ belongs to the Fitting subgroup $F(G)$. We generalize this theorem in terms of certain length parameters of $E_n(g)$. For soluble $G$ we prove that if, for some $n$, the Fitting height of $E_n(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$ the results are in terms of nonsoluble length and gener…
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