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

Abstract

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.

Hyper–(Abelian–by–finite) groups with many subgroups of finite depth

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Soluble groups with many 2-generator torsion-by-nilpotent subgroups

“…Following a question of Erdős, B. H. Neumann proved in [18] that a group is centre-by-finite if, and only if, every infinite subset contains a commuting pair of distinct elements. Since this result, problems of similar nature have been the object of many papers (for example [1]- [7], [10], [15]- [17], [21]- [23]). In particular, in [15] Lennox and Wiegold considered the class (Ω, ∞) of groups in which every infinite subset contains two distinct elements generating an Ω-group, where Ω is a given class of groups.…”

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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.…”