If \(\mathcal{N}\) is a class of groups, then \(L_{n}(\mathcal{N})\) denote the class of all groups \(G\) in which the normal closure \(\langle g_{1}, \ldots , g_{n}\rangle^{G}\) of each \(n\)-generated subgroup \(\langle g_{1}, \ldots, g_{n}\rangle\) belongs to \(\mathcal{N}\). If \(\mathcal{N}\) is a quasivariety, then so is \(L_{n}(\mathcal{N})\) [the author, Sib. Math. J. 60, No. 4, 565–571 (2019; Zbl 1475.08002); translation from Sib. Mat. Zh. 60, No. 4, 724–733 (2019)]. This provides a chain \[ L_{1}(\mathcal{N}) \supseteq L_{2}(\mathcal{N}) \supseteq \dots \supseteq L_{n}(\mathcal{N}) \supseteq \dots \supseteq \mathcal{N} \tag{\(\ast\)}\] in which \(\bigcap_{n=1}^{\infty} L_{n}(\mathcal{N})=\mathcal{N}\). This chain is called infinite if it contains infinitely many different quasivarieties \(L_{n}(\mathcal{N})\).
In this article, the author finds some conditions for chain \((\ast)\) to be infinite. In particular if \(\mathcal{N}\) is generated by a finitely generated nilpotent non-abelian group then \((\ast)\) is infinite.