State and prove Maschke's Theorem for complex representations of finite groups.

Without using character theory, show that every irreducible complex representation of the dihedral group of order $10, D_{10}$, has dimension at most two. List the irreducible complex representations of $D_{10}$ up to isomorphism.

Let $V$ be the set of vertices of a regular pentagon with the usual action of $D_{10}$. Explicitly decompose the permutation representation $\mathbb{C} V$ into a direct sum of irreducible subrepresentations.