On the complexity of the lattice of quasivarieties of nilpotent groups

Abstract

Let $p$ be a prime number. Denote by $\mathfrak{R}_{\delta, \lambda}$ the non-abelian variety
of nilpotent groups of class at most 2 of exponent $p^\delta$ with commutator subgroup of exponent $p^\lambda;$
by $F_2$ the free goup of rank 2 in $\mathfrak{R}_{\delta, \lambda};$ by $qH$ the quasivariety of groups
generated by a group $H.$ It is proved that the interval $[qF_2, qG]$ is continual if all
the following conditions are true: $G\in\mathfrak{R}_{\delta, \lambda},$ $G$ is a finite group
defined in $\mathfrak{R}_{\delta, \lambda}$ by commutator defining relations, $qF_2\varsubsetneq qG.$