Groups which satisfy a Thue–Morse identity

2019 ◽  
Vol 19 (10) ◽  
pp. 2050191
Author(s):  
H. Khosravi ◽  
A. Faramarzi Salles
Keyword(s):  
Semigroup Identity ◽  
Engel Group ◽  
Locally Nilpotent

In this paper, we study 3-Thue–Morse groups, but these are the groups satisfying the semigroup identity [Formula: see text]. We prove that if [Formula: see text] is a 3-Thue–Morse group then [Formula: see text] is soluble for every [Formula: see text] and [Formula: see text] in [Formula: see text]. Furthermore, if [Formula: see text] is an Engel group without involution then we show that [Formula: see text] is locally nilpotent.

scholarly journals On (𝑛 + ½)-Engel groups

2020 ◽  
Vol 23 (3) ◽  
pp. 503-511
Author(s):  
Enrico Jabara ◽  
Gunnar Traustason
Keyword(s):  
Positive Integer ◽  
Engel Group ◽  
Locally Nilpotent ◽  
The Law

AbstractLet n be a positive integer. We say that a group G is an {(n+\frac{1}{2})}-Engel group if it satisfies the law {[x,{}_{n}y,x]=1}. The variety of {(n+\frac{1}{2})}-Engel groups lies between the varieties of n-Engel groups and {(n+1)}-Engel groups. In this paper, we study these groups, and in particular, we prove that all {(4+\frac{1}{2})}-Engel {\{2,3\}}-groups are locally nilpotent. We also show that if G is a {(4+\frac{1}{2})}-Engel p-group, where {p\geq 5} is a prime, then {G^{p}} is locally nilpotent.

A NOTE ON THE LOCAL NILPOTENCE OF 4-ENGEL GROUPS

2005 ◽  
Vol 15 (04) ◽  
pp. 757-764 ◽  
Author(s):  
GUNNAR TRAUSTASON
Keyword(s):  
Engel Group ◽  
Locally Nilpotent

Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth–Bendix algorithm. In this paper we give a short handproof of the nilpotency of T.

GROUP ALGEBRAS WHOSE UNIT GROUP IS LOCALLY NILPOTENT

2020 ◽  
Vol 109 (1) ◽  
pp. 17-23 ◽  
Author(s):  
V. BOVDI
Keyword(s):  
Group Algebra ◽  
Unit Group ◽  
Complete List ◽  
Group Algebras ◽  
Engel Group ◽  
Nilpotent Elements ◽  
Locally Nilpotent ◽  
Normalized Units

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 set of nontrivial nilpotent elements of $FG$ is finite and nonempty, and $V(FG)$ is an Engel group.

GROUP ALGEBRAS WITH ENGEL UNIT GROUPS

2016 ◽  
Vol 101 (2) ◽  
pp. 244-252 ◽  
Author(s):  
M. RAMEZAN-NASSAB
Keyword(s):  
Normal Subgroup ◽  
Group Algebra ◽  
Group Algebras ◽  
Abelian Normal Subgroup ◽  
Locally Finite ◽  
Engel Group ◽  
Group Of Units ◽  
Nilpotent Elements ◽  
Locally Nilpotent ◽  
Formula Group

Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$-group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$-group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$.

scholarly journals A note on Engel groups and local nilpotence

1998 ◽  
Vol 64 (1) ◽  
pp. 92-100 ◽  
Author(s):  
R. G. Burns ◽  
Yuri Medvedev
Keyword(s):  
Group Theory ◽  
Large Class ◽  
Finite Groups ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Residually Finite ◽  
Engel Group ◽  
Residually Finite Groups ◽  
Locally Nilpotent

AbstractThis paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

TWO GENERATOR 4-ENGEL GROUPS

2005 ◽  
Vol 15 (02) ◽  
pp. 309-316 ◽  
Author(s):  
GUNNAR TRAUSTASON
Keyword(s):  
Engel Group ◽  
Locally Nilpotent

Using known results on 4-Engel groups one can see that a 4-Engel group is locally nilpotent if and only if all its 3-generator subgroups are nilpotent. As a step towards settling the question whether all 4-Engel groups are locally nilpotent we show that all 2-generator 4-Engel groups are nilpotent.

Engel groups with an identity

2019 ◽  
Vol 29 (01) ◽  
pp. 1-7
Author(s):  
Pavel Shumyatsky ◽  
Antonio Tortora ◽  
Maria Tota
Keyword(s):  
Finite Group ◽  
Affirmative Answer ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Engel Group ◽  
Engel Elements ◽  
Locally Nilpotent

We give an affirmative answer to the question whether a residually finite Engel group satisfying an identity is locally nilpotent. More generally, for a residually finite group [Formula: see text] with an identity, we prove that the set of right Engel elements of [Formula: see text] is contained in the Hirsch–Plotkin radical of [Formula: see text]. Given an arbitrary word [Formula: see text], we also show that the class of all groups [Formula: see text] in which the [Formula: see text]-values are right [Formula: see text]-Engel and [Formula: see text] is locally nilpotent is a variety.

On 4-Engel Groups

2007 ◽  
Vol 10 ◽  
pp. 341-353 ◽  
Author(s):  
Michael Vaughan-Lee
Keyword(s):  
Normal Closure ◽  
Engel Group

In this note, the author proves that a group G is a 4-Engel group if and only if the normal closure of every element g ∈ G is a 3-Engel group

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