Suppose that $\left(\rho_{1}, V_{1}\right)$ and $\left(\rho_{2}, V_{2}\right)$ are complex representations of the finite groups $G_{1}$ and $G_{2}$ respectively. Use $\rho_{1}$ and $\rho_{2}$ to construct a representation $\rho_{1} \otimes \rho_{2}$ of $G_{1} \times G_{2}$ on $V_{1} \otimes V_{2}$ and show that its character satisfies

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

for each $g_{1} \in G_{1}, g_{2} \in G_{2}$.

Prove that if $\rho_{1}$ and $\rho_{2}$ are irreducible then $\rho_{1} \otimes \rho_{2}$ is irreducible as a representation of $G_{1} \times G_{2}$. Moreover, show that every irreducible complex representation of $G_{1} \times G_{2}$ arises in this way.

Is it true that every complex representation of $G_{1} \times G_{2}$ is of the form $\rho_{1} \otimes \rho_{2}$ with $\rho_{i}$ a complex representation of $G_{i}$ for $i=1,2 ?$ Justify your answer.