On the dimension subgroups of metabelian groups

Abstract Let G be a finitely presented group given by its pre-abelian presentation X 1 ,…, X m ; X e 1 1 ζ 1 ,…, X e m m ζ,ζ m +1,…>, where e i ≥0 for i = 1,…, m and ζ j ϵ G ′ for j ≥1. Let N be the subgroup of G generated by the normal subgroups [ x e i i , G ] for i = 1,…, m . Then D n +2 ( G )≡γ n +2 ( G ) (mod NG ′) for all n ≥0, where G ” is the second commutator subgroup of G ,γ n +2 ( G ) is the ( n +2)th term of the lower central series of G and D n +2 ( G ) = G ∩(1+△ n +2 ( G )) is the ( n +2)th dimension subgroup of G .