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On Metanilpotent Varieties of Groups
Published online by Cambridge University Press: 20 November 2018
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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.
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