Metabelian groups and varieties

Abstract

The principal concern of this paper is varieties of metabelian groups, and we obtain generalizations of results of Cossey (1966), Brisley (1967) and Weichsel (1967) with classifications of certain metabelian varieties of finite exponent. Examples are given to show that the lattice of metabelian varieties is not distributive. To do these things requires a discussion of varieties of certain universal algebras closely related to groups, called split-groups, which arise in a natural way from non-nilpotent metabelian critical groups. Work by Brooks (1968) and L. G. Kovacs & M. F. Newman (unpublished) make it seem likely that in the near future a complete classification of all non-nilpotent, join-irreducible, metabelian varieties will be obtained. For varieties of certain split-groups closely related to metabelian groups we obtain such a description.