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It can be shown that there exists a homomorphism between $SU(2)$ and $SO(3)$. What is the physical implication of a homomorphism actually? I know that in physics $SU(2)$ acts on a space of spin 1/2 particle whereas $SO(3)$ is a group of rotations in Euclidean space. Now that there is a homomorphism between the two groups, what does it imply?

Moreover, my text only present the case of homomorphism between $SU(2)$ and $SO(3)$, but what about a general $SU(n)$ with $SO(3)$? Is there also homomorphism between them?

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