(a) Define the group $S^{1}$. Sketch a proof of the classification of the irreducible continuous representations of $S^{1}$. Show directly that the characters obey an orthogonality relation.

(b) Define the group $S U(2)$.

(i) Show that there is a bijection between the conjugacy classes in $G=S U(2)$ and the subset $[-1,1]$ of the real line. [If you use facts about a maximal torus $T$, you should prove them.]

(ii) Write $\mathcal{O}_{x}$ for the conjugacy class indexed by an element $x$, where $-1<x<1$. Show that $\mathcal{O}_{x}$ is homeomorphic to $S^{2}$. [Hint: First show that $\mathcal{O}_{x}$ is in bijection with $G / T$.

(iii) Let $t: G \rightarrow[-1,1]$ be the parametrisation of conjugacy classes from part (i). Determine the representation of $G$ whose character is the function $g \mapsto 8 t(g)^{3}$.