Lecture 3 – Small bias with respect to linear tests

1. χtrivial(x) = 1 for every x ∈ G is a character. It is called the trivial character. 2. If χ1 and χ2 are characters of G then so is χ1 · χ2 (where χ1 · χ2(x) = χ1(x)χ2(x)). 3. For every character χ of G and x ∈ G, |χ(x)| = 1 (the absolute norm is of course in C). In particular, if we define χ1(x) = χ1(x), then χ1 is also a character and χ · χ = χtrivial. 4. This implies that Ĝ = {χ ∈ C[G] | χ is a character} is an Abelian group, with identity as in item (1), multiplication as in item (2) and inverse as in item (3).