A relation between the index of a fundamental group and the covering map

Question

My professor just enunciated this statement:

$|\pi_1(X,x_0):\pi_1(\tilde X,\tilde x_0)|=\#P^{-1}(x_0)$

where $P$ is the covering $P:\tilde X\rightarrow X$, such that $P(\tilde x_0)=x_0$.

I've tried for hours to prove this, but I can't get it done. I would appreciate any help.

Thanks in advance.

Answer 1

Take a class $[\alpha] \in \pi_1(X,x_0)$. How many lifts of $\alpha$ exist in $\pi_1(\tilde{X},\tilde{x_0})$?

This might be useful https://en.wikipedia.org/wiki/Covering_space#Lifting_properties