What is this group $O(2)/\mathbb{Z}_2$?

Question

I'm talking from a physics student perspective. What is the properties of this group?. Is it isomorphic to something more known in the physics community?.

Background of the question:

I found this group while playing around with the properties of $O(2)$ group of ortogonal matrices. At first sight, I thought that $O(2)/\mathbb{Z}_2$ was isomorphic to $SO(2)$, but I think it's wrong.

Answer 1

Interestingly, the group $PO(2, \Bbb R)$ is isomorphic to $O(2, \Bbb R)$.

To see this, let us write $r(\theta)$ for $\begin{pmatrix}\cos \theta & \sin\theta \\-\sin\theta&\cos\theta\end{pmatrix}$ and write $t(\theta)$ for $\begin{pmatrix}\cos \theta & \sin\theta \\\sin\theta&-\cos\theta\end{pmatrix}$.

We then have: \begin{eqnarray}r(\theta)r(\theta') &=& r(\theta + \theta'),\\r(\theta)t(\theta') &=& t(\theta' - \theta),\\t(\theta)r(\theta') &=& t(\theta + \theta'),\\t(\theta)t(\theta') &=& r(\theta' - \theta).\end{eqnarray}

Now define a map $\phi:O(2, \Bbb R) \rightarrow O(2, \Bbb R)$ sending $r(\theta)$ to $r(2\theta)$ and $t(\theta)$ to $t(2\theta)$. From the above formulas, it is clear that $\phi$ is a surjective (topological) group homomorphism.

The kernel of $\phi$ is clearly the subgroup $\{\pm I_2\}$, thus $\phi$ induces an isomorphism from $PO(2, \Bbb R)$ to $O(2, \Bbb R)$.