Algorithmic use of the Mal'cev correspondence

Mal'cev showed in the 1950s that there is a correspondence between radicable torsion-free nilpotent groups and rational nilpotent Lie algebras. In this paper we show how to establish the connection between the radicable hull of a finitely generated torsion-free nilpotent group and its corresponding Lie algebra algorithmically. We apply it to fast multiplication of elements of polycyclically presented groups

Testing polycyclicity of finitely generated rational matrix groups

We describe algorithms for testing polycyclicity and nilpotency for finitely generated subgroups of GL( d, Q) and thus we show that these properties are decidable. Variations of our algorithm can be used for testing virtual polycyclicity and virtual nilpotency for finitely generated subgroups of GL(d, Q).</p