(i) Let $H$ be a normal subgroup of the group $G$. Let $G / H$ denote the group of cosets $\tilde{g}=g H$ for $g \in G$. If $D: G \rightarrow G L\left(\mathbb{C}^{n}\right)$ is a representation of $G$ with $D\left(h_{1}\right)=D\left(h_{2}\right)$ for all $h_{1}, h_{2} \in H$ show that $\tilde{D}(\tilde{g})=D(g)$ is well-defined and that it is a representation of $G / H$. Show further that $\tilde{D}(\tilde{g})$ is irreducible if and only if $D(g)$ is irreducible.

(ii) For a matrix $U \in S U(2)$ define the linear map $\Phi_{U}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ by $\Phi_{U}(\mathbf{x}) \cdot \boldsymbol{\sigma}=$ $U \mathbf{x} . \boldsymbol{\sigma} U^{\dagger}$ with $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)^{T}$ as the vector of the Pauli spin matrices

$$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

Show that $\left\|\Phi_{U}(\mathbf{x})\right\|=\|\mathbf{x}\|$. Because of the linearity of $\Phi_{U}$ there exists a matrix $R(U)$ such that $\Phi_{U}(\mathbf{x})=R(U) \mathbf{x}$. Given that any $S U(2)$ matrix can be written as

$$U=\cos \alpha I-i \sin \alpha \mathbf{n} \cdot \boldsymbol{\sigma}$$

where $\alpha \in[0, \pi]$ and $\mathbf{n}$ is a unit vector, deduce that $R(U) \in S O(3)$ for all $U \in S U(2)$. Compute $R(U) \mathbf{n}$ and $R(U) \mathbf{x}$ in the case that $\mathbf{x} \cdot \mathbf{n}=0$ and deduce that $R(U)$ is the matrix of a rotation about $\mathbf{n}$ with angle $2 \alpha$.

[Hint: $\mathbf{m} . \boldsymbol{\sigma} \mathbf{n} . \boldsymbol{\sigma}=\mathbf{m} . \mathbf{n} I+i(\mathbf{m} \times \mathbf{n}) . \boldsymbol{\sigma} .]$

Show that $R(U)$ defines a surjective homomorphism $\Theta: S U(2) \rightarrow S O(3)$ and find the kernel of $\Theta$.