Q:

Is the quotient space O(3)/SO(2) a group?

I can't figure out how to demonstrate if it is a group or not. I should see if O(3) is a simple group or that SO(2) is a normal subgroup of O(3).

A:

Isn't $SO(3)/SO(2)$ as a quotient space just the $2$-sphere $S^2$?
But $S^2$ has no topological group structure
(any orientation-preserving map from $S^2$ to $S^2$ has a fixed point). Then $O(3)/SO(2)$
would be two disjoint $S^2$s and so won't have a topological group structure.

