Let $S^n$ be the $n$-dimensional sphere with $n \geq 2$, and let $G$ be a finite group that acts freely on $S^n$. Suppose $G$ is non-trivial. Then,
(i) Compute the homotopy groups of the quotient space $\pi_i\left(S^n / G\right)$ for $0 \leq i \leq n$.
(ii) Suppose $n$ is even. Prove that $G$ is isomorphic to $\mathbb{Z}_2$.
(iii) Suppose $n$ is odd. Show that $G$ cannot be isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ for $p$ a prime number.