An interesting subarea of group theory deals with the possible approximations of groups by (almost) representations to the finite symmetric groups or finite dimensional unitary groups. A major (still open) problem is whether every group can be approximated by almost homomorphisms to the symmetric groups.

Group stability offers a way to attack such problems. We will observe that this is a form of property testing and will show some success (and some work in progress) to apply ideas from the previous lectures to this problem.