On Metanilpotent Varieties of Groups

Abstract

Let denote the variety of all groups which are extensions of a nilpotent-of-class- c group by a nilpotent-of-class- d group, and let denote the variety of all metabelian groups. The main result of this paper is the following theorem. THEOREM. Let be a subvariety of which does not contain . Then every -group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent. In particular, a finitely generated torsion-free -group is a nilpotent-by-finite group. This generalizes the main theorem of Ŝmel′kin [ 4 ], where the same result is proved for subvarieties of , where is the variety of abelian groups. See also Lewin and Lewin [ 2 ] for a related discussion.