Publ. Math. Debrecen 67/1-2 (2005), 93–102 Soluble groups with many 2-generator torsion-by-nilpotent subgroups By NADIR TRABELSI (S´etif) Abstract. We prove in this paper that a finitely generated soluble group in which every infinite subset contains a pair of distinct elements x, y such that x, y is torsion-by-nilpotent (respectively, x, x y  is Chernikov-by-nilpotent), is itself torsion-by-nilpotent (respectively, finite-by-nilpotent). 1. Introduction and results Following a question of Erd˝ os, 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 simi- lar nature have been the object of many papers (for example [1]–[7], [10], [15]–[17], [21]–[23]). In particular, in [15] Lennox and Wiegold con- sidered 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. They characterised finitely generated soluble groups which be- long to (Ω, ∞) when Ω is the class of polycyclic, or nilpotent, or coherent groups. Here we will consider the class (Ω, ∞), when Ω is the class TN of torsion-by-nilpotent groups, or the class CN of Chernikov-by-nilpotent groups, and we will prove the following results: Mathematics Subject Classification: 20F16. Key words and phrases: finitely generated soluble groups, finite-by-nilpotent groups, infinite subsets, Chernikov groups, torsion groups. I would like to thank the referee whose comments improved the exposition of this paper.