Let $G$ be a finite group, and suppose $G$ acts on the finite sets $X_{1}, X_{2}$. Define the permutation representation $\rho_{X_{1}}$ corresponding to the action of $G$ on $X_{1}$, and compute its character $\pi_{X_{1}}$. State and prove "Burnside's Lemma".

Let $G$ act on $X_{1} \times X_{2}$ via the usual diagonal action. Prove that the character inner product $\left\langle\pi_{X_{1}}, \pi_{X_{2}}\right\rangle$ is equal to the number of $G$-orbits on $X_{1} \times X_{2}$.

Hence, or otherwise, show that the general linear group $\mathrm{GL}_{2}(q)$ of invertible $2 \times 2$ matrices over the finite field of $q$ elements has an irreducible complex representation of dimension equal to $q$.

Let $S_{n}$ be the symmetric group acting on the set $X=\{1,2, \ldots, n\}$. Denote by $Z$ the set of all 2-element subsets $\{i, j\}(i \neq j)$ of elements of $X$, with the natural action of $S_{n}$. If $n \geqslant 4$, decompose $\pi_{Z}$ into irreducible complex representations, and determine the dimension of each irreducible constituent. What can you say when $n=3$ ?