Galois correspondence for coverings

Remarks

The first clause is the lifting criterion applied to $p_1$ and $p_2$. For the third clause, a loop $\gamma$ at $x_0$ lifts to a path from $\tilde x_0$ to a point $\tilde x_1$ of the fibre. A deck transformation sending $\tilde x_0$ to $\tilde x_1$ exists exactly when $[\gamma]$ normalises $H_p$, and it is unique, which gives a surjection $N(H_p) \to G(p)$ with kernel $H_p$. For the covering $S^1 \to S^1$, $z \mapsto z^n$, the subgroup is $n\Z \subseteq \Z$, which is normal, and the deck group is $\Z / n\Z$, acting by rotations through multiples of $2\pi / n$. Together with the classification of covering spaces, these facts form a dictionary between coverings and subgroups.

Going deeper

For a student

The universal cover plays the part of the algebraic closure, and the fundamental group plays the part of the absolute Galois group. Normal coverings correspond to Galois extensions, and the deck group corresponds to the Galois group. The analogy becomes an exact theorem in Grothendieck's theory of the étale fundamental group, where finite étale coverings replace topological coverings. In practice, the statement computes deck groups and tells which quotients of $\pi_1(X)$ arise as symmetry groups of coverings.