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How can I prove that the quotient group $S^3/\{+I,-I\}$ is isomorphic to $SO_3$ and that the group $S^3$ is not isomophic to $SO_3$?

Here $S^3$ is the subgroup of the quaternion group: $S^3=\{a+bi+cj+dk | a^2+b^2+c^2+d^2=1\}$

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    What do you mean by $S^3/\{\pm I\}$? Are you looking at the orbit space of $S^3$ under the action of $\Bbb{Z}/2\Bbb{Z}$ given by the antipodal map?2012-11-26

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