Improving tameness for metabelian groups

We show that any finitely generated metabelian group can be embedded in a metabelian group of type F₃. More generally, we prove that if n is a positive integer and Q is a finitely generated abelian group, then any finitely generated ℤQ-module can be embedded in a module that is n-tame. Combining with standard facts, the F₃ embedding theorem follows from this and a recent theorem of R. Bieri and J. Harlander.