I'm trying to solve the following problem.
Let $S^1$ denote the set of $z \in \mathbb{C}^{*}$ with modulus $1$ and $H = \{\pm 1\}$ a subgroup. Prove that $S^1/H \cong S^1$.
I'm a bit stumped with how to prove this. The first thing that came to mind was the first isomorphism theorem. In that case, I'd want to write down a surjective homomorphism $\phi: S^1 \to S^1$ with kernel $H$. As $H$ is not the trivial subgroup, this homomorphism can't be an isomorphism, so the homomorphisms that originally came to mind, such as the identity homomorphism or the authomorphism map, will not work. I then thought to use a quotient map, but the canonical quotient map would send an element of $S^1$ to the coset it's a member of: it'd be surjective, but not necessarily injective.
I'd appreciate any help with this problem and some direction on whether any of my above ideas are on the right track.
Define $\phi: S^1 \to S^1$, $$\phi(z)=z^2$$ for all $z\in S^{1}.$
It remains to show that $\phi$ is a surjective homomorphism with kernel $H$.