Let \(\mathfrak{M}\) be the class of groups satisfying the minimal condition on normal subgroups and let Ω be the class of groups of finite lower central depth, that is groups G such that γi(G) = γi+1(G) 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 ∈ G, there exists a normal subgroup Hx of finite index in G satisfying \(\langle x,x^{h}\rangle\in\mathfrak{M}\Omega\) for every h ∈ Hx, then G is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated hyper-(Abelian-by-finite) group G such that for every x ∈ G, there exists a normal subgroup Hx of finite index in G satisfying \(\langle x,x^{h}\rangle\in\mathfrak{T}\Omega\) for every h ∈ Hx, is periodic-by-nilpotent; where \(\mathfrak{T}\) stands for the class of periodic groups.
