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 clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

scholarly journals Almost Engel finite and profinite groups

2016 ◽  
Vol 26 (05) ◽  
pp. 973-983 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Profinite Group ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Nilpotent Residual ◽  
A Finite Group

Let [Formula: see text] be an element of a group [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] be the subgroup generated by all commutators [Formula: see text] over [Formula: see text], where [Formula: see text] is repeated [Formula: see text] times. We prove that if [Formula: see text] is a profinite group such that for every [Formula: see text] there is [Formula: see text] such that [Formula: see text] is finite, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] 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 [Formula: see text], we prove that if, for some [Formula: see text], [Formula: see text] for all [Formula: see text], then the order of the nilpotent residual [Formula: see text] is bounded in terms of [Formula: see text].

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (1) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

Some Engel conditions on infinite subsets of certain groups

2000 ◽  
Vol 62 (1) ◽  
pp. 141-148 ◽  
Author(s):  
Alireza Abdollahi
Keyword(s):  
Positive Integer ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Metabelian Group ◽  
Infinite Subset ◽  
Central Series ◽  
Finitely Generated ◽  
Locally Nilpotent ◽  
Finite Quotient

Let k be a positive integer. We denote by ɛk(∞) the class of all groups in which every infinite subset contains two distinct elements x, y such that [x,k y] = 1. We say that a group G is an -group provided that whenever X, Y are infinite subsets of G, there exists x ∈ X, y ∈ Y such that [x,k y] = 1. Here we prove that:(1) If G is a finitely generated soluble group, then G ∈ ɛ3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3.(2) If G is a finitely generated metabelian group, then G ∈ ɛk(∞) if and only if G/Zk (G) is finite, where Zk (G) is the (k + 1)-th term of the upper central series of G.(3) If G is a finitely generated soluble ɛk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt (G) is finite.(4) If G is an infinite -group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G 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.

scholarly journals Locally nilpotent skew linear groups

1986 ◽  
Vol 29 (1) ◽  
pp. 101-113 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Positive Integer ◽  
Division Ring ◽  
Matrix Ring ◽  
Linear Groups ◽  
Full Matrix ◽  
Locally Nilpotent ◽  
Completely Reducible ◽  
Full Matrix Ring

Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.

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.

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.

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.

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