# S0(3)/SO(2) is isomorphic to the projective plane

• eddo
In summary, for the given assignment, the goal is to show that S0(3)/SO(2) is isomorphic to the projective plane. However, there seems to be some confusion as a textbook claims that SO(3)/SO(2) is isomorphic to the 2-sphere. It is clarified that both statements are correct, as S^n \cong SO(n+1)/SO(n) and \mathbb{R}P^n \cong SO(n+1)/O(n). Therefore, SO(3)/SO(2) is isomorphic to the 2-sphere and SO(3)/O(2) is isomorphic to the projective plane. This makes sense intuitively due

#### eddo

For an assignment, my prof asked that we show that S0(3)/SO(2) is isomorphic to the projective plane (ie the 2-sphere with antipodal points identified). Here's my problem. I checked in a textbook for some help, and it claimed that SO(3)/SO(2) is isomorphic to the 2-sphere. So which one is right? It would be nice to know what I should be trying to prove before I put too much work in. Thank you.

eddo said:
For an assignment, my prof asked that we show that S0(3)/SO(2) is isomorphic to the projective plane (ie the 2-sphere with antipodal points identified). Here's my problem. I checked in a textbook for some help, and it claimed that SO(3)/SO(2) is isomorphic to the 2-sphere. So which one is right? It would be nice to know what I should be trying to prove before I put too much work in. Thank you.

$$S^n \cong SO \left( n+1 \right)/SO \left( n \right)$$

and

$$\mathbb{R}P^n \cong SO \left( n+1 \right)/O \left( n \right),$$

so $SO \left( 3 \right)/SO \left( 2 \right)$ is isomorphic to the 2-sphere, and $SO \left( 3 \right)/O \left( 2 \right)$ is isomorphic to the projective plane.

Last edited:
Thank you. Intuitively that's what I expected to be the case, because of the relationship between O(n) and SO(n). It makes sense that antipodal points get identified if you mod out by O(n) because the reflections that O(n) has over SO(n) would identify these points. But you can't always count on intuition so thank you for verifying this.