Let A and B be two permutations in Sym(n) that “almost commute” — are they a small deformation of permutations that truly commute? More generally, if R is a system of words-equations in variables X={x1,…,xn} and A1,…,An are permutations that are nearly solutions; are they near true solutions?

It turns out that the answer to this question depends only on the group presented by the generators X and relations R. This leads to the notions of “stable groups” and “testable groups”.

We will present a few results and methods which were developed in recent years to check whether a group is stable or testable. We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic, and with invariant random subgroups.