5 - Nilpotent Groups, Commutators and Nilprogressions

Summary

We introduce nilpotent groups, define nilprogressions and nilpotent progressions, and present some of their basic properties. We start by introducing the Heisenberg group. We present some specific examples of nilprogressions and nilpotent progressions in the Heisenberg group and show that they have small tripling. We then define general nilpotent groups and present their basic properties. Next, we introduce commutators, the collecting process and basic commutators. Finally, we define nilprogressions and nilpotent progressions in general, show that they have small tripling and show that the notions of nilprogression and nilpotent progression are roughly equivalent.