Anastasia Hadjievangelou, Marialaura Noce, Gunnar Traustason
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
Let m,n be positive integers, v a multilinear commutator word and w=v^m. We prove that if G is an orderable group in which all w-values are n-Engel, then the verbal subgroup v(G) is locally nilpotent. We also show that in the particular case where v=x the group G is nilpotent (rather than merely locally nilpotent).
Lie methods for countably categorical Engel groups: the Wilson conjecture for $4$-Engel $5$-groups
该研究运用李方法，结合计算机代数系统，证明了可数范畴的3-Engel群、4-Engel 5-群等满足Wilson猜想相关结论，还探讨了三类代数结构间幂零性结果的转移。
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.
5-Engel Lie algebras
该研究证明了特征零或素特征p>7域上的5-Engel李代数幂零类至多11，同时证明p>7的有限5-Engel p-群幂零类至多10，推进了Engel代数与群的幂零性研究。
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 $G$.
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 prove that if a linear group $G$ is almost Engel, then $G$ is finite-by-hypercentral. If $G$ is almost nil, then $G$ is finite-by-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 $G$ 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.
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.
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…
Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.
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 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 $q$ 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.
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。
Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.
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…
We say that a group $G$ is almost Engel if for every $g\in G$ there is a finite set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$, that is, for every $x\in G$ there is a positive integer $n(x,g)$ such that $[...[[x,g],g],\dots ,g]\in {\mathscr E}(g)$ if $g$ is repeated at least $n(x,g)$ times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose ${\mathscr E}(g)=\{ 1\}$ for all $g\in G$.) We prove that if a compact (Hausdorff) group $G$ is almost Engel, then $G$ has a finite no…
A subset $S$ of a group $G$ is called an Engel set if, for all $x,y\in S$, there is a non-negative integer $n=n(x,y)$ such that $[x,\,_n y]=1$. In this paper we are interested in finding conditions for a group generated by a finite Engel set to be nilpotent. In particular, we focus our investigation on groups generated by an Engel set of size two.
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 $G$.
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.
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。
Let $ n, q $ be positive integers. We show that if $ G $ is a finitely generated residually finite group satisfying the identity $ [x,_ny^q]\equiv 1, $ then there exists a function $ f(n) $ such that $ G $ has a nilpotent subgroup of finite index of class at most $ f(n) $. We also extend this result to locally graded groups.
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 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…
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.
Let R be an associative ring with unity and let [R] and U(R) denote the associated Lie ring (with [a,b]=ab-ba) and the group of units of R, respectively. In 1983 Gupta and Levin proved that if [R] is a nilpotent Lie ring of class c then U(R) is a nilpotent group of class at most c. The aim of the present note is to show that, in general, a similar statement does not hold if [R] is n-Engel. We construct an algebra R over a field of characteristic different from 2 and 3 such that the Lie algebra [R] is 5-Engel but U(R) is not a 5-Engel group.
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 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.
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 $G$.
Wilson conjecture for omega-categorical Lie algebras, the case 3-Engel characteristic 5
本文证明了特征5域上满足特定恒等式的ω-范畴3-Engel李代数是幂零的，并扩展介绍了Wilson猜想及其变体与局部和全局幂零问题的联系。
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.
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…
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…
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…
On Zel'manov's global nilpotence theorem for Engel Lie algebras
本文给出特征零域上n-Engel李代数的Zel'manov全局幂零定理的证明，该定理推广了Kostrikin的局部幂零结论，注记旨在梳理Zel'manov的原创论证思路，供其他研究者参考。
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。
A group word $w$ is said to be strongly concise in a class $\mathscr C$ of profinite groups if, for any group $G$ in $\mathscr C$, either $w$ takes at least continuum values in $G$ or the verbal subgroup $w(G)$ is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups. In the present paper we prove that for any…
Here we show that a finite nilpotent group is 2-closed if and only if it is either cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.
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…
Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are n-Engel and w(G) is locally nilpotent is a variety (Theorem A). Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n-Engel and v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result …
We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an Engel group.
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.
In this paper, we work on the pro-nilpotent group topology of a free group. First we investigate the closure of the product of finitely many subgroups of a free group in the pro-nilpotent group topology. We present an algorithm for the calculation of the closure in the pro-nilpotent group topology of the product of finitely many finitely generated subgroups of a free group. Then we deduce that the nil-closure of a rational subset is computable. We also prove that the pseudovarieties V malcev G_nil, for every decidable pseudovariety of monoids V, and J * G_nil are decidable.
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.
In the paper we characterize the class of finite solvable groups by two-variable identities in a way similar to the characterization of finite nilpotent groups by Engel identities. More precisely, a sequence of words $u_1,...,u_n,... $ is called correct if $u_k\equiv 1$ in a group $G$ implies $u_m\equiv 1$ in a group $G$ for all $m>k$. We are looking for an explicit correct sequence of words $u_1(x,y),...,u_n(x,y),...$ such that a group $G$ is solvable if and only if for some $n$ the word $u_n$ is an identity in $G$. Let $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. The main resu…
Let $G$ be a permutation group on a set $Ω$, and $k$ a positive integer. The $k$-closure $G^{(k)}$ of $G$ is the largest subgroup of $\operatorname{Sym}(Ω)$, with the same as $G$ orbits of componentwise action on $Ω^k$. We prove that the $k$-closure of a finite nilpotent permutation group is the direct product of $k$-closures of its Sylow subgroups.
We study some properties of the coset poset associated with the family of subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is $2$-Engel, and $2$-connected if and only if the group is nilpotent of class $2$ or less. We determine the homotopy type of the coset poset for the group of $4\times 4$ upper unitriangular matrices over $\mathbb{F}_p$, and for the Burnside groups of exponent $3$.
We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.
In this note we investigate the normality of closures of orthogonal and symplectic nilpotent orbits in positive characteristic. We prove that the closure of such a nilpotent orbit is normal provided that neither type d nor type e minimal irreducible degeneration occurs in the closure, and conversely if the closure is normal, then any type e minimal irreducible degeneration does not occur in it. Here, the minimal irreducible degenerations of a nilpotent orbit are introduced by W. Hesselink in [6] (or see [11] from which we take Table 1 for the complete list of all minimal irreducible degenerati…
We associate a graph ${\mathcal N}_{S}$ with a semigroup $S$ (called the upper non-nilpotent graph of $S$). The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). In case $S$ is a group this graph has been introduced by A. Abdollahi and M. Zarrin and some remarkable properties have been proved. The aim of this paper is to study this graph (and some related graphs, such as the non-commuting graph) and to discover the algebraic structure of $S$ determined by the associated graph. It is shown th…
We deal with the following conjecture. If w is a group word and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only. We show that this is true in the case where w is either the nth Engel word or the word [x^n,y_1,y_2,...,y_k] (Theorem A). Further, we show that for any positive integer e there exists a number k=k(e) such that if w is a word and G is a finite group in which any nilpotent subgroup generated by products of k values of the word w has exponent dividing e, …
Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D
本文研究 A、B、D 型约化群中二阶幂零元中心化子在旗簇内的轨道闭包，证明其正规性、特定条件下的 Cohen–Macaulay 性与有理奇点，并构造涉及 Schubert 簇的双有理有理消解。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。