Non-Abelian Homomorphism Testing, and Distributions Close to Their Self-convolutions

In this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism. For two finite groups (not necessarily Abelian), an arbitrary map , and a parameter , say that is -close to a homomorphism if there is some homomorphism such that and differ on at most elements of , and say that is -far otherwise. For a given and , a homomorphism tester should distinguish whether is a homomorphism, or if is -far from a homomorphism. When is Abelian, it was known that the test which picks ! "# %$ random pairs & !' and tests that