State and prove Maschke's theorem.

Let $G$ be the group of isometries of $\mathbb{Z}$. Recall that $G$ is generated by the elements $t, s$ where $t(n)=n+1$ and $s(n)=-n$ for $n \in \mathbb{Z}$.

Show that every non-faithful finite-dimensional complex representation of $G$ is a direct sum of subrepresentations of dimension at most two.

Write down a finite-dimensional complex representation of the group $(\mathbb{Z},+)$ that is not a direct sum of one-dimensional subrepresentations. Hence, or otherwise, find a finitedimensional complex representation of $G$ that is not a direct sum of subrepresentations of dimension at most two. Briefly justify your answer.

[Hint: You may assume that any non-trivial normal subgroup of $G$ contains an element of the form $t^{m}$ for some $m>0$.]