Artin’s theorem

Abstract

Let G be a finite group and let x₁, ..., x _h be its distinct irreducible characters. A class function on G is a character if and only if it is a linear combination of the x _i ’s with non-negative integer coefficients. We will denote by R⁺ (G) the set of these functions, and by R(G) the group generated by R⁺(G), i.e., the set of differences of two characters. We have

$$ R\left( G \right) = {Z_{x1}} \oplus \cdots \oplus Z{x_h}. $$