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 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”.

scholarly journals FINITELY GENERATED SOLUBLE GROUPS WITH A CONDITION ON INFINITE SUBSETS

2012 ◽  
Vol 87 (1) ◽  
pp. 152-157
Author(s):  
ASADOLLAH FARAMARZI SALLES
Keyword(s):  
Soluble Group ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Infinite Set

AbstractLet G be a group. We say that G∈𝒯(∞) provided that every infinite set of elements of G contains three distinct elements x,y,z such that x≠y,[x,y,z]=1=[y,z,x]=[z,x,y]. We use this to show that for a finitely generated soluble group G, G/Z2(G) is finite if and only if G∈𝒯(∞).

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 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.

scholarly journals Periodic groups with many permutable subgroups

1992 ◽  
Vol 53 (1) ◽  
pp. 116-119
Author(s):  
Patrizia Longobardi ◽  
Mercede Maj ◽  
Akbar Rhemtulla ◽  
Howard Smith
Keyword(s):  
Finitely Generated ◽  
Locally Finite ◽  
Finitely Generated Group ◽  
Periodic Groups ◽  
Permutable Subgroups ◽  
Infinite Set

AbstractGroups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.

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 Groups with many permutable subgroups

1990 ◽  
Vol 48 (3) ◽  
pp. 397-401 ◽  
Author(s):  
Mario Curzio ◽  
John Lennox ◽  
Akbar Rhemtulla ◽  
James Wiegold
Keyword(s):  
Free Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Cyclic Subgroups ◽  
Prove Theorem ◽  
Permutable Subgroups ◽  
Infinite Set ◽  
Torsion Free

AbstractWe consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

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.

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 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.

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