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I'm trying to come up with covering spaces of $X= S^1 \lor S^1 \lor S^1$ corresponding to two subgroups of $\pi_1(X)= F_3=\langle a,b,c \rangle$. First, the subgroup $H=\langle a^2, b^2, c^2 \rangle$ and second, the normal subgroup $N= \langle \langle a^2, b^2, c^2 \rangle \rangle$ (the smallest normal subgroup containing $H$).

I've got some candidate covers in mind, but I'm not sure if I'm correct. Below is my idea for the cover corresponding to $N$.

This cover shown is normal, since the map swapping the vertices is a covering space transformation, and so the subgroups corresponding to this cover is normal in $F_3$, and it contains $H$.

If this is the correct cover, how can I show this corresponds to the smallest normal subgroup containing $H$? What about the cover corresponding to $H$ itself? No finite degree cover has worked, and I can't seem to cook up an infinite cover that works.

Any hints would be appreciated!

enter image description here

user10039910
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    The number of vertices is equal to the number of cosets of the subgroup. In both cases, the index of your subgroup is infinite, so the corresponding cover has infinitely many vertices. – Hempelicious Jan 06 '19 at 00:46

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