Class Functions on a Group

Definition: Let $G$ be a group. A Class Function on $G$ is a function $\varphi : G \to \mathbb{C}$ with the property that for all $g \in G$ and all $h \in G$ we have that $\varphi(g) = \varphi(hgh^{-1})$.

As we have already noted - if $V$ is a representation of $G$ then the character $\chi_V$ is a class function since:

(1)

\begin{align} \quad \chi_V(g) = \chi_V(hgh^{-1}) \end{align}