How do I prove that every covering map of $S^1$ is isomorphic to one of the following covering maps? $$\varepsilon: \mathbb{R} \to S^1, \quad z\mapsto e^{2\pi i z} \\ p_n:S^1 \to S^1, \quad z \mapsto z^n$$
My idea: If $q:E \to S^1$ is some covering map, it induces some subgroup of $\mathbb{Z}$ which is either trivial or of the form $n \mathbb{Z}$ (right?).
If it's trivial, then $q$ is isomorphic to $\varepsilon$ by the isomorphism criterion. What if it's of the form $n \mathbb{Z}$ ? What is the induced subgroup of $p_n$?
I think you have already answered your question by yourself. If you consider the subgroup $n \mathbb{Z}$ then, by the criterion you mentioned, you covering has to be isomorphic to $p_n$, since the image of this map is indeed $n \mathbb{Z}$.