Abstract:

$L_n(\mathcal{N})$ be the class of all groups $G$ in which the normal closure of each $n$-generated subgroup of $G$ belongs to $\mathcal{N}$. Let $\mathcal{N}$ be the quasivariety of nilpotent groups of class at most 3 without elements of orders 2 and 5 in which the quasi-identity ${(\forall x)(\forall y)([x,y,x]=1\rightarrow [x,y,y]=1)}$ is true.

In this paper we prove that any group $G$ belonging to the class ${L_1(\mathcal{N})}$ is 3-Engel. In particular, the result is true for the quasivariety generated by the free 2-generated nilpotent group of class at most 3.