Irreducible Algebraic Sets in Metabelian Groups

Abstract

We present the construction for a u-product G1 ○ G2 of two u-groups G1 and G2, and prove that G1 ○ G2 is also a u-group and that every u-group, which contains G1 and G2 as subgroups and is generated by these, is a homomorphic image of G1 ○ G2. It is stated that if G is a u-group then the coordinate group of an affine space Gn is equal to G ○ Fn, where Fn is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks.