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

Published online by Cambridge University Press:  17 April 2009

Bijan Taeri
Bijan Taeri
Affiliation:
Department of Mathematics, Isfahan University of Technology, Isfahan and Institute for Studies in Theoretical Physics and Mathematics, Iran, e-mail: b.taeri@cc.iut.ac.ir
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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 k(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 ∈ k(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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 245 - 254
Copyright
Copyright © Australian Mathematical Society 2001

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