We state Breuillard, Green and Tao’s rough classification of the finite approximate subgroups of an arbitrary group. This states that a finite approximate subgroup of an arbitrary group is contained in a union of a few cosets of a finite-by-nilpotent group, the nilpotent quotient of which has bounded step. We define coset nilprogressions, and show how to deduce a more detailed version of the Breuillard–Green–Tao theorem in which the approximate subgroup is contained in a union of a few translates of a coset nilprogression of bounded rank and step.

We present various applications of Breuillard, Green and Tao’s rough classification of finite approximate groups to groups of polynomial growth. We define polynomial, exponential and intermediate growth, and show that these concepts are stable under changes of generating set and passing to subgroups of finite index. We prove Breuillard, Green and Tao’s result that if a ball of large enough radius in a Cayley graph is of size polynomial in the radius then the underlying group is virtually nilpotent. We deduce that all larger balls also have polynomial bounds on their sizes. We guide the reader in the exercises to Breuillard and Tointon’s results that a finite group of large diameter admits large virtually nilpotent and virtually abelian quotients. We also prove the same authors’ result that a finite simple group has diameter bounded by a small power of the size of the group. We prove an isoperimetric inequality for finite groups due to Breuillard, Green and Tao. Finally, we give a brief high-level introduction to applications of approximate groups to the construction of expanders.