Scientific journal "The News of Altai State University". 2009 #1(61). Mathematics and Mechanics.

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2009

$Mathematics and Mechanics$

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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.

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