Consider the two groups $G_{1}=mathrm{SO}(3)$ and $G_{2}=mathrm{U}(1)$ and a surjective mapping $mu: G_{1} ightarrow G_{2}$. Discuss

Consider the two groups $G_{1}=\mathrm{SO}(3)$ and $G_{2}=\mathrm{U}(1)$ and a surjective mapping $\mu: G_{1} \rightarrow G_{2}$. Discuss whether, in general, $\mu$ is a homomorphism, or not.