isomorphism between subset of SU(2) and SO(3)

Question

I know that there is a surjective map $\Phi : SU(2)\to SO(3) $.

My question is if there is a subgroup $A \subset SU(2)$ such that $\Phi_{|A}:A\to SO(3)$ can be a (group) isomorphism.

What would $A$ be?

Thank you

Answer 1

Such a subgroup $A\subset\operatorname{SU}(2)$ fails to exist on topological grounds; I'll assume it to be known that $\operatorname{SU}(2)\cong S_3$ and $\operatorname{SO}(3)\cong\Bbb{RP}^3$. The usual surjective map $$\Phi:\ \operatorname{SU}(2)\ \longrightarrow\ \operatorname{SO}(3)$$ is a homomorphism of topological groups, hence its restriction to $A$ is a homeomorphism. This implies that $S_3$ contains a subspace homeomorphic to $\Bbb{RP}^3$, a contradiction.

Answer 2

Hint If there were such a subgroup $A$, then for any $g \in SU(2) - A$, $A \cup gA$ would be a separation of $SU(2)$---but $SU(2)$ is connected.