scholarly journals A conjecture of Lennox and Wiegold concerning supersoluble groups

1983 ◽  
Vol 35 (2) ◽  
pp. 218-220 ◽  
Author(s):  
J. R. J. Groves
Keyword(s):  
Soluble Group ◽  
Infinite Subset ◽  
Supersoluble Group ◽  
Finitely Generated ◽  
Supersoluble Groups

AbstractWe prove a conjecture of Lennox and Wiegold that a finitely generated soluble group, in which every infinite subset contains two elements generating a supersoluble group, is finite-by-supersoluble.

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.

scholarly journals Lower central depth in finitely generated soluble-by-finite groups

1978 ◽  
Vol 19 (2) ◽  
pp. 153-154 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Depth ◽  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated ◽  
Central Depth

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

scholarly journals Extensions of a problem of Paul Erdös on groups

1981 ◽  
Vol 31 (4) ◽  
pp. 459-463 ◽  
Author(s):  
John C. Lennox ◽  
James Wiegold
Keyword(s):  
Soluble Group ◽  
Finitely Generated ◽  
Infinite Set

AbstractThe main results are as follows. A finitely generated soluble group G is polycyclic if and only if every infinite set of elements of G contains a pair generating a polycyclic subgroup; and the same result with “polycyclic” replaced by “coherent”.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

scholarly journals Locally graded groups with a nilpotency condition on infinite subsets

2000 ◽  
Vol 69 (3) ◽  
pp. 415-420 ◽  
Author(s):  
Costantino Delizia ◽  
Akbar Rhemtulla ◽  
Howard Smith
Keyword(s):  
Positive Integer ◽  
Infinite Subset ◽  
Nilpotent Subgroup ◽  
Finitely Generated ◽  
Finite Image ◽  
Nontrivial Subgroup ◽  
Locally Graded Groups

AbstractA group G is locally graded if every finitely generated nontrivial subgroup of G has a nontrivial finite image. Let N (2, k)* denote the class of groups in which every infinite subset contains a pair of elements that generate a nilpotent subgroup of class at most k. We show that if G is a finitely generated locally graded N (2, k)*-group, then there is a positive integer c depending only on k such that G/Zc (G) is finite.

scholarly journals Characterisation of nilpotent-by-finite groups

2000 ◽  
Vol 61 (1) ◽  
pp. 33-38 ◽  
Author(s):  
Nadir Trabelsi
Keyword(s):  
Positive Integer ◽  
Finite Groups ◽  
Soluble Group ◽  
Finitely Generated ◽  
Positive Integers

LetGbe a finitely generated soluble group. The main result of this note is to prove thatGis nilpotent-by-finite if, and only if, for every pairX,Yof infinite subsets ofG, there exist anxinX,yinYand two positive integersm=m(x,y),n=n(x,y) satisfying [x,nym] = 1. We prove also that ifGis infinite and ifmis a positive integer, thenGis nilpotent-by-(finite of exponent dividingm) if, and only if, for every pairX,Yof infinite subsets ofG, there exist anxinX,yinYand a positive integern=n(x,y) satisfying [x,nym] = 1.

Probability and Bias in Generating Supersoluble Groups

2015 ◽  
Vol 59 (4) ◽  
pp. 899-909 ◽  
Author(s):  
Eleonora Crestani ◽  
Giovanni De Franceschi ◽  
Andrea Lucchini
Keyword(s):  
Uniform Distribution ◽  
Supersoluble Group ◽  
Supersoluble Groups ◽  
Generating Set ◽  
Minimum Integer

AbstractWe discuss some questions related to the generation of supersoluble groups. First we prove that the number of elements needed to generate a finite supersoluble groupGwith good probability can be quite a lot larger than the smallest cardinality d(G) of a generating set ofG. Indeed, ifGis the free prosupersoluble group of rankd⩾ 2 and dP(G) is the minimum integerksuch that the probability of generatingGwithkelements is positive, then dP(G) = 2d+ 1. In contrast to this, ifk–d(G) ⩾ 3, then the distribution of the first component in ak-tuple chosen uniformly in the set of all thek-tuples generatingGis not too far from the uniform distribution.

scholarly journals Groups covered by finitely many nilpotent subgroups

1994 ◽  
Vol 50 (3) ◽  
pp. 459-464 ◽  
Author(s):  
Gérard Endimioni
Keyword(s):  
Soluble Group ◽  
Finitely Generated ◽  
Finite Covering ◽  
Infinite Set

Let G be a finitely generated soluble group. Lennox and Wiegold have proved that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that (x, y) is nilpotent. The main theorem of this paper is an improvement of the previous result: we show that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that [x, ny] = 1 for some integer n = n(x, y) ≥ 0.

scholarly journals Permutable word products in groups

1989 ◽  
Vol 40 (2) ◽  
pp. 243-254 ◽  
Author(s):  
P.S. Kim ◽  
A.H. Rhemtulla
Keyword(s):  
Soluble Group ◽  
Study Groups ◽  
Finitely Generated

Let u(x1,…,xn) = x11 … x1m be a word in the alphabet x1, …,xn such that x1i ≠ x1i for all i = 1,…, m − 1. If (H1, …, Hn) is an n-tuple of subgroups of a group G then denote by u(H1, …, Hn) the set {u(h1,…,hn) | hi ∈ Hi}. If σ ∈ Sn then denote by uσ(H1,…,Hn) the set u(Hσ(1),…,Hσ(n)). We study groups G with the property that for each n-tuple (H1,…,Hn) of subgroups of G, there is some σ ∈ Sn σ ≠ 1 such that u(H1,…,Hn) = uσ(H1,…,Hn). If G is a finitely generated soluble group then G has this property for some word u if and only if G is nilpotent-by-finite. In the paper we also look at some specific words u and study the properties of the associated groups.

CHARACTERISTIC RELATIONS FOR A FINITE-BY-NILPOTENT GROUP

2005 ◽  
Vol 15 (02) ◽  
pp. 273-277
Author(s):  
GÉRARD ENDIMIONI
Keyword(s):  
Positive Integer ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Finitely Generated

We improve previous results by showing that a finitely generated soluble group G is finite-by-nilpotent if and only if for all a, b ∈ G, there exists a positive integer n such that [a, nb] belongs to γn+2(<a, b>).<a<a,b>>′.

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