Embedding Properties of Metabelian Lie Algebras and Metabelian Discrete Groups

We show that for every natural number m a finitely generatedmetabelian group G embeds in a quotient of a metabelian groupof type FP_m. Furthermore, if m 4, the group G can be embeddedin a metabelian group of type FP_m. For L a finitely generatedmetabelian Lie algebra over a field K and a natural number mwe show that, provided the characteristic p of K is 0 or p >m, then L can be embedded in a metabelian Lie algebra of typeFP_m. This result is the best possible as for 0 < p m everymetabelian Lie algebra over K of type FP_m is finite dimensionalas a vector space.