I know that $SU(2)$ is diffeomorphic to 3-sphere $S^3$. And, I know that $U(2)/U(1)$ is diffeomorphic to $S^3$. I would like to know if there is an Lie group isomorphism between $U(2)/U(1)$ and $SU(2)$ in the sense that it is a diffeomorphic (bijective and smooth) and it is a group homomorphism.
Appreciate.
No, they are not isomorphic. The group $U(2)/U(1)$ is (assuming that you see $U(1)$ as $\{e^{i\theta}\operatorname{Id}\mid\theta\in\Bbb R\}$) the projective unitary group $PU(2)$, which is isomoprphic to $SU(2)/\{\pm\operatorname{Id}\}$, but not it $SU(2)$.