Suppose that $G$ is a finite group. Define the inner product of two complex-valued class functions on $G$. Prove that the characters of the irreducible representations of $G$ form an orthonormal basis for the space of complex-valued class functions.

Suppose that $p$ is a prime and $\mathbb{F}_{p}$ is the field of $p$ elements. Let $G=\mathrm{GL}_{2}\left(\mathbb{F}_{p}\right)$. List the conjugacy classes of $G$.

Let $G$ act naturally on the set of lines in the space $\mathbb{F}_{p}^{2}$. Compute the corresponding permutation character and show that it is reducible. Decompose this character as a sum of two irreducible characters.