Groupes ayant beaucoup de sous-groupes 2-engendrés dans une classe donnée

Abstract

Given a class of groups , we denote by ( ,∞)* the class of groups G in which every infinite subset contains two distinct elements x,y such that 〈x,xy⟩Î . We also denote by F the class of groups G such that for every element x in G there exists a normal subgroup H(x) of finite index in G satisfying 〈x,h⟩Î for every h Î H(x). In this thesis, we have shown that a finitely generated hyper-(Abelianby- finite) group G is finite-by-nilpotent if, and only if, G is in the class ( ,∞)* (respectively F ), where is either the class of groups of finite depth, or the class of extensions of a group satisfying the minimal condition on normal subgroups by an Engel group. We have also shown that any finitely generated hyper-(Abelian-by-finite) group in F is in , where is respectively the class of nilpotent-by-finite, finite-by-nilpotent and periodic-by-nilpotent groups