Normal covering space and resulting isomorphisms When studying for an exam, I came across this question in Hatcher, and it has stumped me. 
Let $G$ be a group with normal subgroup $N$. Show that there exists a normal covering space $\tilde{X} \rightarrow X$ with $\pi_1(X) \approx G, \pi_1(\tilde{X}) \approx N$, and deck transformation group $G(\tilde{X}) \approx G/N$.
I've been reading about classification theorems but haven't yielded much progress with those. The results in section 1.3 of Hatcher seem to rely on assumptions of path-connectedness, which we don't have here. Help?

As you say, the standard results relating covering spaces and subgroups of the fundamental group tend to require some hypotheses about the space you are talking about.  However, here you aren't given a space $X$; you're allowed to choose $X$ to be whatever space you want!  So you can just pick $X$ to be some nice space with fundamental group $G$, and then the usual results apply.  For instance, you could take $X$ to be a connected CW-complex with fundamental group $G$, and then the covering of $X$ associated to the subgroup $N$ will have the desired properties.  Such a CW-complex can be constructed using a presentation of $G$ (take a wedge of circles with one circle for each generator, and then attach a 2-cell for each relation); details can be found in Corollary 1.28 of Hatcher.
