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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).

Let $𝔐$ be the class of groups satisfying the minimal condition on normal subgroups and let $\Omega$ be the class of groups of finite lower central depth, that is groups $G$ such that ${\gamma }_i\left(G\right)={\gamma }_{i+1}\left(G\right)$ for some positive integer $i$. The main result states that if $G$ is a finitely generated hyper-(Abelian-by-finite) group such that for every $x\in G$, there exists a normal subgroup $H_x$ of finite index in $G$ satisfying $\langle x,x^h\rangle \in 𝔐\Omega$ for every $h\in H_x$, then $G$ is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated hyper-(Abelian-by-finite)...