Let $G=\mathrm{SU}(2)$.

(i) Sketch a proof that there is an isomorphism of topological groups $G /\{\pm I\} \cong$ $\mathrm{SO}(3)$

(ii) Let $V_{2}$ be the irreducible complex representation of $G$ of dimension 3. Compute the character of the (symmetric power) representation $S^{n}\left(V_{2}\right)$ of $G$ for any $n \geqslant 0$. Show that the dimension of the space of invariants $\left(S^{n}\left(V_{2}\right)\right)^{G}$, meaning the subspace of $S^{n}\left(V_{2}\right)$ where $G$ acts trivially, is 1 for $n$ even and 0 for $n$ odd. [Hint: You may find it helpful to restrict to the unit circle subgroup $S^{1} \leqslant G$. The irreducible characters of $G$ may be quoted without proof.]

Using the fact that $V_{2}$ yields the standard 3-dimensional representation of $\mathrm{SO}(3)$, show that $\bigoplus_{n \geqslant 0} S^{n} V_{2} \cong \mathbb{C}[x, y, z]$. Deduce that the ring of complex polynomials in three variables $x, y, z$ which are invariant under the action of $\mathrm{SO}(3)$ is a polynomial ring in one generator. Find a generator for this polynomial ring.