V.V. Lodeyshchikova

On Levi Quasivarieties Generated by Nilpotent Groups

For an arbitrary class *M* of groups, denote by *L* (*M*) a class of all groups *G* for which the normal closure of any element belongs to *M*; *qM* is the quasivariety generated by *M*.

In this paper we prove that if *K* is an arbitrary class of torsion-free nilpotent groups of class ≤2 containing a non-abelian group and such that in every group of K the centralizer of each element outside the center of the group is an abelian subgroup, then *L* (*qM*) coincides with the quasivariety of torsion-free nilpotent groups of class ≤3.

If *K* is an arbitrary class of nilpotent groups of class ≤2 of exponent *p* (*p* is a prime number, *p*≠2) containing a non-abelian group and such that in every group of K the centralizer of each element outside the center of the group is an abelian subgroup, then *L* (*qM*) coincides with the variety of nilpotent groups of class ≤3 of exponent *p*.

*Keywords*: quasivariety, Levi classes, nilpotent groups.