Levi Classes of Quasivarieties of Nilpotent Groups of Exponent ps

Abstract

The Levi class L(M) generated by the class M of groups is the class of all groups in which the normal closure of every element belongs to M. It is proved that there exists a set of quasivarieties M of cardinality continuum such that \( L\left(\mathrm{M}\right)=L\left(q{H}_{p^s}\right) \), where \( q{H}_{p^s} \) is the quasivariety generated by the group \( {H}_{p^s} \), a free group of rank 2 in the variety \( {R}^{p^s} \) of ≤ 2-step nilpotent groups of exponent p^s with commutator subgroup of exponent p, p is a prime number, p ≠ 2, s is a natural number, s ≥ 2, and s > 2 for p = 3.