Lecture Notes on Linearity (Group Homomorphism) Testing

Let G and H be two groups. For simplicity, we denote by + the group operation in each of these groups. A function f : G → H is called a (group) homomorphism if for every x, y ∈ G it holds that f(x+ y) = f(x) + f(y). One important special case of interest is when H is a finite field and G is a vector space over this field; that is, G = H for some natural number m. In this case, a homomorphism f from G to H can be presented as f(x1, ..., xm) = ∑m i=1 cixi, where x1, ..., xm, c1, ..., cm ∈ H. In this case, f is a linear function over H, which explains why testing group homomorphism is often referred to as linearity testing. Group homomorphisms are among the simplest and most basic classes of finite functions. They may indeed claim the title of the most natural algebraic functions. This chapter addresses the problem of testing whether a given function is a group homomorphism or is far from any group homomorphism.