LEVI CLASSES GENERATED BY NILPOTENT GROUPS

Let L(M) be a class of all groups G for which the normal closure (x)^G of every element x belongs to a class M. L(M) is a Levi class generated by M. Let N and N(0) be classes of finitely generated nilpotent groups and of torsion-free. Finitely generated. nilpotent groups. respectively. We prove that qN(o) (belong to) L(qN(o)) and qN (belong to) L(qN(o)). and so L(qN(o)) # qL(N(o)) and L(qN) # qL(N).It is shown that quasivarieties L(qN) and L(qN(o)) are closed under free products. and that each contains at most one maximal proper subquasivariety. It is also proved that L(M) is closed under free products if so is M.