Given a covering space action of a group \(G\) on a path-connected, locally path-connected space \(X\), then each subgroup \(H \subset G\) determines a composition of covering spaces \(X \to X/H \to X/G\). Show:
Every path-connected covering space between \(X\) and \(X/G\) is isomorphic to \(X/H\) for some subgroup \(H \subset G\).
Two such covering spaces \(X/H_1\) and \(X/H_2\) of \(X/G\) are isomorphic if \(H_1\) and \(H_2\) are conjugate subgroups of \(G\).
The covering space \(X/H \to X/G\) is normal if \(H\) is a normal subgroup of \(G\), in which case the group of deck transformations of this cover is \(G/H\).
Solution
(a).
1. Let \(p : Y \to X/G\) be a path-connected covering space, and choose a basepoint \(y_0 \in Y\) with \(p(y_0) = x_0\).
setup.
2. Since \(X \to X/G\) is the universal cover (a covering space action of \(G\) on a simply-connected \(X\)), there is a lift \(\tilde p : X \to Y\) of the covering \(X \to X/G\) through \(p\).
the universal property of the universal cover (lifting criterion).
3. The deck transformation group of \(X \to X/G\) is \(G\), and the subgroup \(H = p_*(\pi_1(Y))\) (or equivalently the stabilizer of the lift) acts on \(X\) with quotient \(Y\).
covering space theory; \(Y \cong X/H\) where \(H\) is the subgroup of \(G\) corresponding to the subgroup \(p_*(\pi_1(Y)) \le \pi_1(X/G) = G\).
4. Hence \(Y \cong X/H\) for some subgroup \(H \le G\).
3.
(b).
1. \(X/H_1\) and \(X/H_2\) are isomorphic as covering spaces of \(X/G\) iff \(H_1\) and \(H_2\) are conjugate in \(G\).
two subgroups give isomorphic covers iff they are conjugate (the isomorphism is induced by a deck transformation of \(X\) conjugating one subgroup to the other).
2. Hence the claim.
1.
(c).
1. The cover \(X/H \to X/G\) is normal iff \(H\) is normal in \(G\).
a covering is normal iff the corresponding subgroup is normal in the fundamental group.
2. If \(H \trianglelefteq G\), the deck transformation group of \(X/H \to X/G\) is \(N_G(H)/H = G/H\).
the deck transformations of \(X/H \to X/G\) are the elements of \(G\) normalizing \(H\), modulo \(H\); when \(H\) is normal, \(N_G(H) = G\).
3. Q.E.D.
4 (a), 2 (b), 2 (c).