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

Le principal résultat de cet article est qu’un groupe $G$ hyper-(Abélien-par-fini) de type fini est fini-par-nilpotent si, et seulement si, toute partie infinie de $G$ contient deux éléments distincts $x,y$ tels que ${\gamma }_n\left(〈x,x^y〉\right)={\gamma }_{n+1}\left(〈x,x^y〉\right)$ pour un certain entier positif $n=n(x,y)$ (respectivement, $〈x,x^y〉$ est une extension d’un groupe vérifiant la condition minimale sur les sous-groupes normaux par un groupe d’Engel).

The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group $G$ is finite-by-nilpotent if and only if every infinite subset contains two distinct elements $x$, $y$ such that ${\gamma }_n\left(〈x\text{,}{x}^y〉\right)$ $={\gamma }_{n+1}\left(〈x\text{,}{x}^y〉\right)$ for some positive integer $n=n(x,y)$ (respectively, $〈x,x^y〉$ is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group).

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