Probably a very easy question but I can't seem to figure it out: What is the group $\frac{U(1)}{\mathbb{Z_2}}$ isomorphic to?
My intuition is telling me to use the homomorphism theorem, so I want to find a homomorphism $\phi:U(1) \to Im(\phi)$ with $Ker(\phi)=\{-1,1\}$.
My first thought was to define $\phi(z)=z^2,\forall z \in U(1)$ since then we get $Ker(\phi)=\{-1,1\}$.
However, if I'm not mistaken, this results in $\frac{U(1)}{\mathbb{Z_2}}\cong U(1)$. What went wrong? Thanks in advance for your help!