If $\rho_{1}: G_{1} \rightarrow G L\left(V_{1}\right)$ and $\rho_{2}: G_{2} \rightarrow G L\left(V_{2}\right)$ are representations of the finite groups $G_{1}$ and $G_{2}$ respectively, define the tensor product $\rho_{1} \otimes \rho_{2}$ as a representation of the group $G_{1} \times G_{2}$ and show that its character is given by

$$\chi_{\rho_{1} \otimes \rho_{2}}\left(g_{1}, g_{2}\right)=\chi_{\rho_{1}}\left(g_{1}\right) \chi_{\rho_{2}}\left(g_{2}\right)$$

Prove that

(a) if $\rho_{1}$ and $\rho_{2}$ are irreducible, then $\rho_{1} \otimes \rho_{2}$ is an irreducible representation of $G_{1} \times G_{2}$;

(b) each irreducible representation of $G_{1} \times G_{2}$ is equivalent to a representation $\rho_{1} \otimes \rho_{2}$ where each $\rho_{i}$ is irreducible $(i=1,2) .$

Is every representation of $G_{1} \times G_{2}$ the tensor product of a representation of $G_{1}$ and a representation of $G_{2}$ ?