If $X$ is path-connected and has a finite fundamental group then every covering map from $X$ to $X$ is an homeomorphism.

Question

If $X$ is path-connected and has a finite fundamental group then every covering map from $X$ to $X$ is an homeomorphism. I am trying to prove that the covering space has just one sheet since the number of elements in the fundamental group is related with the sheets. But this tell me I have a finite number of sheets: how do I prove I only have one?

Answer 1

Hint: If $p : (Y, y_0) \to (X, x_0)$ is a covering map, then $p_\star : \pi_1(Y, y_0) \to \pi_1(X, x_0)$ is injective. (Hatcher Prop. 1.31)

Furthermore, the index $[\pi_1(X, x_0) : p_\star(\pi_1(Y, y_0))]$ is the number of sheets of the covering map $p$. (Hatcher Prop. 1.32)

Now you can think about what happens if $X = Y$ and $\pi_1(X, x_0)$ is finite...