What is $\mathbb{S}^{1}/\{\pm {1},\pm {i}\}$ isomorphic to

Question

I am starting to self-study Lie Groups, and came across this question.

What is $\mathbb{S}^{1}/\{\pm {1},\pm {i}\}$ isomorphic to

To begin with, it seems as if $\{\pm {1},\pm {i}\}$ are actually only two points - so I am wondering if I am correct in this, and if so, what is the rationale for this?

I can show that $\{\pm {1},\pm {i}\}$ is a normal subgroup of $\mathbb{S}^{1}$, so it is the kernel of a group homomorphism. As such, since the questions asks for an isomorphism, these elements are sent to the identity element.

And that's about as far as I get. Thanks.

Answer 1

Hint: Consider $S^1\to S^ 1$, $z\mapsto z^4$.