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

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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 $O(2)$.

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    $\begingroup$ The center of $SU(2)$ is $\pm 1$; its only nontrivial quotient is $SO(3)$. $\endgroup$ May 28, 2015 at 17:59

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