State Maschke's Theorem for finite-dimensional complex representations of the finite group $G$. Show by means of an example that the requirement that $G$ be finite is indispensable.

Now let $G$ be a (possibly infinite) group and let $H$ be a normal subgroup of finite index $r$ in $G$. Let $g_{1}, \ldots, g_{r}$ be representatives of the cosets of $H$ in $G$. Suppose that $V$ is a finite-dimensional completely reducible $\mathbb{C} G$-module. Show that

(i) if $U$ is a $\mathbb{C} H$-submodule of $V$ and $g \in G$, then the set $g U=\{g u: u \in U\}$ is a $\mathbb{C} H$-submodule of $V$;

(ii) if $U$ is a $\mathbb{C} H$-submodule of $V$, then $\sum_{i=1}^{r} g_{i} U$ is a $\mathbb{C} G$-submodule of $V$;

(iii) $V$ is completely reducible regarded as a $\mathbb{C} H$-module.

Hence deduce that if $\chi$ is an irreducible character of the finite group $G$ then all the constituents of $\chi_{H}$ have the same degree.