I am having problems with 1.3.24 in Hatcher's Algebraic Topology:
24. Given a covering space action of a group G on a path-connected, locally path-connected space X, then each subgroup H in G determines a composition of covering spaces X -> X/H -> G. Show:
a. Every path-connected covering space between X and X/G is isomorphic to X/H for some subgroup H in G
The best I can do here is say that since we have a covering space action then pi_1(X/H1) = pi_1(X/H2); not sure how to proceed.
b. Two such covering spaces X/H1 and H/H2 of X/G are isomorphic iff H1 and H2 are conjugate subgroups of G.
c. The covering space X/H -> X/G is normal iff H is a normal subgroup of G, in which case the group of deck transformations of this cover is G/H.
(definition of covering space actionat (*) on page 72 in this pdf http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf)