Levi Classes of Quasivarieties of Nilpotent Groups of Class at Most Two

Abstract

A Levi class \(L\left(\mathcal{M}\right)\) generated by a class \(\left(\mathcal{M}\right)\) of groups is the class of all groups in which the normal closure of every cyclic subgroup belongs to \(\left(\mathcal{M}\right)\). Let p be a prime and p ≠ 2, let H_p be a free group of rank 2 in the variety of nilpotent groups of class at most 2 with commutator subgroup of exponent p, and let qH_p be the quasivariety generated by the group H_p. It is shown that there exists a set of quasivarieties \(\mathcal{M}\) of cardinality continuum such that \(L\left(\mathcal{M}\right)\) = L(qH_p). Let s be a natural number, s ≥ 2. We specify a system of quasi-identities defining L(q(H_p, \({Z}_{{p}^{s}}\))), and prove that there exists a set of quasivarieties \(\mathcal{M}\) of cardinality continuum such that \(L\left(\mathcal{M}\right)\) = L(q(H_p, \({Z}_{{p}^{s}}\))), where \({Z}_{{p}^{s}}\) is a cyclic group of order p^s; q(H_p, \({Z}_{{p}^{s}}\)) is the quasivariety generated by the groups H_p and \({Z}_{{p}^{s}}.\)