Abstract:

In this paper we find a condition under which the completion of a torsion-free nilpotent group  from the variety $\mathcal{N}_3$ of nilpotent groups of class $\leq 3$ is contained in the quasivariety  generated by this group. We show that the completion of each group from a quasivariety $qF(\mathcal{N}_3)$ generated by the free $3$-nilpotent group $F(\mathcal{N}_3)$ belongs to $qF(\mathcal{N}_3)$.  We  prove the theorem which allows us to extract roots from a commutant of  groups from $qF(\mathcal{N}_3)$ while remaining in $qF(\mathcal{N}_3)$. We find conditions under which the $\mathcal{N}_3$-free product of groups from $qF(\mathcal{N}_3)$ is again contained in $qF(\mathcal{N}_3)$. We show that   $qF(\mathcal{N}_3)$ is not closed with respect to $\mathcal{N}_3$-free products.