Looking for a non trivial homomorphism I

Question

Is there a non trivial homomorphism $f: SU(2) \to O(2)$? Is there a concrete description of $Hom(SU(2), O(2))$?

Answer 1

No. $\text{ker}(f)$ is a normal subgroup of $SU(2)$. Since $SU(2)$ is a simple Lie group, its normal subgroups are either trivial subgroup or its center or itself. So $\text{im}(f)=SU(2)$ or $SU(2)/\{\pm I\}$, or the trivial group. Among them only the trivial group is a subgroup of $O(2)$.