3. For $G_1$ and $G_2$ finite groups and $V_1$, $V_2$ finite dimensional complex representations of $G_1$, $G_2$ respectively, the tensor product $V_1 \otimes V_2$ is given an action of $G_1 \times G_2$ by
$(g_1, g_2). (v_1 \otimes v_2) = (g_1.v_1) \otimes (g_2.v_2).$
(a) Show that this is a representation.
(b) Calculate its character in terms of $\chi_{V_1}$ and $\chi_{V_2}$.
(c) If $V_1$, $V_2$ are irreducible, show that $V_1 \otimes V_2$ is irreducible.
(d) Show that all irreducible representations of $G_1 \times G_2$ arise this way.
