pointwise product of two characters of $G$ is a character of $G$ Let $\phi: G \to GL_{n}(\mathbb{C})$ and $\rho: G \to GL_{m}(\mathbb{C})$ be representations. Let $V = M_{mn}(\mathbb{C})$. Define the representations $\tau : G \to GL(V)$ by $\tau_{g}(A) = \rho_{g}A\phi_{g}^{T}$.
I know that $\chi_{\tau}(g) = \chi_{\rho}(g)\chi_{\psi}(g) \ \forall \ g \in G$. What is the best argument to show that the pointwise product of two characters of $G$ is a character of $G$?  

Have you learned about the tensor product of two representations?  The character of $\phi\otimes\rho$ is the pointwise product of the characters of $\phi$ and $\rho$.
