We are given a cover of the figure eight-
Show that $\pi_1$ of this cover is a normal subgroup with index $\mathbb{Z}$.
I have shown that the given cover is regular, hence the image of its fundamental group is normal in $G=\pi_1(S^1 \vee S^1)$. If the red and blue loops are labelled $a$ and $b$, then it seems that the fundamental group of the cover is $H = \langle b^kab^{-k}\rangle$. It seems that its cosets are precisely $b^nH$; I can show that any word in $a,b$ can be written in this form.
But, I can't formalize my ideas.
Edit: I now see that $H=\langle a\rangle$. So, words in $G/H$ all look like $b^n$ which makes the idea of "number of $b$'s" in a given word from $G$ well defined; this number will identify the coset $b^nH$ the word belongs to.