I have shown that the unit quaternions $S^3$ act transitively on $S^2$ via the conjugation map $v\mapsto xvx^{-1}$ and that the stabilizer of $i$ equals $S^1$.
I should be able to get an isomorphism $f: S^3/S^1 \to S^2$, correct? I have tried to define $f$ by $[g]\mapsto gig^{-1}$, but I just can't show that this is a homomorphism. (I think I've shown that it's a bijection.)
Furthermore, this map is supposed to equal the Hopf map given by mapping each $(z,w)\in S^3\subset \mathbb C^2$ to the projective line $[z,w]\in \mathbb C \mathbb P^1$.
Perhaps once I find the map $f$ explicitly, this last point will be clear. But what map $f$ would give the desired isomorphism?
UPDATE: I made a silly mistake, as $S^2$ is not a group. I just want a simple bijection (not an isomorphism), just as I defined.
However, I would still appreciate if someone could explain how we get the Hopf map from this.