The class of finitely generated torsion-free nilpotent groups has proven to play an important role in the study of infinite solvable groups, both in understanding the structure of infinite solvable groups and in the development of algorithms for studying them. Here we describe an algorithm for deciding if a given finitely generated torsion-free nilpotent group is decomposable as the direct product of two non-trivial subgroups, and we show how to compute such a decomposition if it exists. Central to our decidability proof are some new results about the way that the decompositions of a finitely generated torsion-free nilpotent group are related to the decompositions of the group’s rational closure.

This is joint work with Gilbert Baumslag and Chuck Miller.