I'm studying Linear groups and already bamboozled after proving that there is a bijective correspondence from $SU_2$to $S^3$ and $SU_2$ can be thought of as the set of unit vectors in the quaternion algebra! [what could be more interesting than visualizing groups? :-)]
I have luckily encountered two awesome questions-
$1.$ There is no way to make 2-sphere into a group.
$2.$ The only spheres on which one can define CONTINUOUS GROUP LAWS are the 1-sphere and 3-sphere!
I don't know how to prove these results, I know basic algebraic topology, need help to know if these results have simple proofs and please share some articles describing more about the geometry of linear groups.
Thanks in advance!