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: $S^3=\{a+bi+cj+dk | a^2+b^2+c^2+d^2=1\}$
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: $S^3=\{a+bi+cj+dk | a^2+b^2+c^2+d^2=1\}$
To prove that $S^3/\pm 1$ is isomorphic to $SO(3)$, use the hypersphere of rotations. This gives a way of naturally thinking of $SO(3)$ as a quotient of $S^3$. Embedding everything inside the quaternions makes defining the mapping much cleaner.
To show that $SO(3)$ and $S^3$ are not isomorphic as lie groups, note that this gives a homeomorphism of the underlying topological spaces. But from the hypersphere of rotations, one can also see that $SO(3)$ and $\mathbb{RP}^3$ are homeomorphic, and $\pi_{1}(\mathbb{RP}^3) = \mathbb{Z}/2\mathbb{Z}$, whereas $\pi_{1}(S^3) = \{1\}$ since the $3$-sphere is simply connected.