Groups covered by finitely many nilpotent subgroups

Endimioni, Gérard

doi:10.1017/s0004972700013575

Cited by 10 publications

(10 citation statements)

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“…Since this result, problems of similar nature have been the object of many papers (for example [1], [2], [3], [4], [5], [9], [11], [10], [16], [17]). We present here some further results of the same type.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Centre-by-metabelian groups with a condition on infinite subsets

Trabelsi¹

2003

Publicacions Matemàtiques

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In this note, we consider some combinatorial conditions on infinite subsets of groups and we obtain in terms of these conditions some characterizations of the classes L(N k )F and FL(N k ) for the finitely generated centre-by-metabelian groups, where L(N k ) (respectively, F ) denotes the class of groups in which the normal closure of each element is nilpotent of class at most k (respectively, finite groups).

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Section: Introduction and Resultsmentioning

confidence: 99%

Centre-by-metabelian groups with a condition on infinite subsets

Trabelsi¹

2003

Publicacions Matemàtiques

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“…The extension of the questions of Paul Erdos, firstly, is considered by Lennox and Wiegold [15]. Further questions of a similar nature, with different aspects, have been considered by many people (see for example [1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,17,23,24,20,25,26,27]).…”

Section: Introduction and Resultsmentioning

confidence: 99%

A question of Paul Erdös and nilpotent-by-finite groups

Taeri¹

2001

Bull. Austral. Math. Soc.

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Let n be a positive integer or infinity (denoted ∞), k a positive integer. We denote by Ωk(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist distinct elements x, y ∈ X and integers t0, t1…, tk such that , where xi, ∈ {x, y}, i = 0, 1,…,k, x0 ≠ x1. If the integers t0, t1,…,tk are the same for any subset X of G, we say that G is in the class Ω̅k(n). The class k (n) is defined exactly as Ωk(n) with the additional conditions . Let t2, t3,…,tk be fixed integers. We denote by the class of all groups G such that for any infinite subsets X and Y there exist x ∈ X, y ∈ Y such that , where xi ∈ {x, y}, x0 ≠ x1, i = 2, 3, …, k. Here we prove that (1) If G ∈ k(2) is a finitely generated soluble group, then G is nilpotent.(2) If G ∈ Ωk(∞) is a finitely generated soluble group, then G is nilpotentby-finite.(3) If G ∈ Ω̅k(n), n a positive integer, is a finitely generated residually finite group, then G is nilpotent-by-finite.(4) If G is an infinite -group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient, then G is nilpotent-by-finite.

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“…Also note that Endimioni in [8] showed that every finitely generated soluble group belonging to is nilpotent, from which it follows that every finitely generated infinite soluble group belonging to m n is nilpotent, since m n ⊆ . Lemma 2.10.…”

Section: Downloaded By [Stanford University Libraries] At 20:32 16 Ocmentioning

confidence: 99%

Ensuring a Group is Weakly Nilpotent

Zarrin

2012

Communications in Algebra

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Groups covered by finitely many nilpotent subgroups

Cited by 10 publications

References 9 publications

Centre-by-metabelian groups with a condition on infinite subsets

Centre-by-metabelian groups with a condition on infinite subsets

A question of Paul Erdös and nilpotent-by-finite groups

Ensuring a Group is Weakly Nilpotent

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