Let $\displaystyle G$ be a finite subgroup of $\displaystyle \mathbb{C}^*$. For example, $\displaystyle \{ 1,-1,i,-i \}$ is a subgroup of order 4.

(a) Prove that $\displaystyle G$ must be a subset of $\displaystyle T=\{ z \in \mathbb{C}^* | \text{ } |z|=1 \}.$

(b) Prove that $\displaystyle G$ is cyclic and is generated by an element of form $\displaystyle e^{\frac{2 \pi i}{n}}$.