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We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.

If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isomorphism, our ultimate concern. It turns out that many of the groups in a given equivalence class share various properties including finite presentability. There are examples, such as the lamplighter group, where an equivalence class consists of a single isomorphism class and others where this is not the case. We give several examples where we solve the Isomorphism Problem. We prove also that the sequence of torsion-free ranks of the lower central quotients of a finitely generated metabelian group is computable. In a future paper we plan on proving that there is an algorithm to compute the numerator and denominator of the rational Poincar\'e series of a finitely generated metabelian group and will carry out this computation in a number of examples, which may shed a tiny bit of light on the Isomorphism Problem. Our proofs use localization, class field theory and some constructive commutative algebra., Comment: 24 pages. Substantially rewritten. Some new results

In this paper we study the residual nilpotence of groups defined by basic commutators. We prove that the so-called Hydra groups as well as certain of their generalizations and quotients are, in the main, residually torsion-free nilpotent. By way of contrast we give an example of a group defined by two basic commutators which is not residually torsion-free nilpotent., Comment: 17 pages

A group is metabelian if its commutator subgroup is abelian. For finitely generated metabelian groups, classical commutative algebra, algebraic geometry and geometric group theory, especially the latter two subjects, can be brought to bear on their study. The object of this paper is to describe some of the new ideas and open problems that arise., Comment: 17 pages

We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality., Comment: DVI and Post-Script files only. To appear in J. London Math. Soc

The object of this paper is to describe a simple method for proving that certain groups are residually torsion-free nilpotent, to describe some new parafree groups and to raise some new problems in honour of the memory of Wilhelm Magnus., Comment: AMS-Tex, 8 pages, no figures. To appear in Trans. Am. Math. Soc