Define the circle group $U(1)$. Give a complete list of the irreducible representations of $U(1)$

Define the spin group $G=S U(2)$, and explain briefly why it is homeomorphic to the unit 3-sphere in $\mathbb{R}^{4}$. Identify the conjugacy classes of $G$ and describe the classification of the irreducible representations of $G$. Identify the characters afforded by the irreducible representations. You need not give detailed proofs but you should define all the terms you use.

Let $G$ act on the space $\mathrm{M}_{3}(\mathbb{C})$ of $3 \times 3$ complex matrices by conjugation, where $A \in S U(2)$ acts by

$$A: M \mapsto A_{1} M A_{1}^{-1}$$

in which $A_{1}$ denotes the $3 \times 3$ block diagonal matrix $\left(\begin{array}{cc}A & 0 \\ 0 & 1\end{array}\right)$. Show that this gives a representation of $G$ and decompose it into irreducibles.