Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$:
a) $O(3)=SO(3)\otimes \{1,-1\}$
b) $SO(3)\simeq SU(2)/Z_2$
Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If so, is $O(3)\simeq SU(2)$ (both locally and topologically)?
Can anyone help me with my above confusion? Any hints or reference are welcome. Thanks a lot!
They are not the same. $O(3)$ is not connected (they have $det = \pm 1$), while $SU(2) \cong \mathbb S^3$ is connected. However, they are all locally the same, as $O(3)$ is locally the same as $SO(3)$ and $SU(2) \to SO(3)$ is a covering.