TITLE: how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$
QUESTION [28 upvotes]: Could any one give me hint for this one?
how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$,
well, Is it the same: there is a 2-fold covering map from $SU(2)$ to $SO(3)$?
what is that map will be?

REPLY [2 votes]: The connection between SO(3) and SU(2) is particularly well and simply described, using quaternions as Lorban suggested above, in the following article,
https://en.wikipedia.org/wiki/Rotation_group_SO(3)?wteswitched=1#Using_quaternions_of_unit_norm
The 2:1-nature of the map between quaternions and SO(3) results from the use of conjugation, i. e.  p ↦ $q p q^{-1}$ , to describe the rotation of p $\in\Bbb{R}^3$, by q $\in\Bbb{H}$ and the fact that both "q" and "-q" define the same rotation of p.