We present Tointon’s proof of Freiman’s theorem in an arbitrary nilpotent group. More specifically, we show that a finite K-approximate subgroup of a nilpotent group of bounded step is contained in a relatively small coset progression of rank bounded by a polynomial in K. We start by treating the torsion-free case, where the details are easiest. As part of our proof of the general case we show that if X is a union of subgroups in an abelian p-group of rank r then the subgroup generated by X has diameter at most r with respect to X. We also show that if H is a subgroup of a nilpotent group G generated by a K-approximate group A, and H is contained in a bounded power of A, then the normal closure of H in G is also contained in a bounded power of A.

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.