I am trying to understand the details of Allen Hatcher's proof of the Seifert–van Kampen theorem (page 44-6 of Algebraic Topology).
My question is regarding the same part of the proof mentioned in this answer which I copy below for convenience:
In the previous paragraph, Hatcher defines two moves that can be performed on a factorization of $[f]$. The second move is
Regard the term $[f_i]\in\pi_1(A_\alpha)$ as lying in the group $\pi_1(A_\beta)$ rather than $\pi_1(A_\alpha)$ if $f_i$ is a loop in $A_\alpha\cap A_\beta$.
Regarding this move, Hatcher asserts that
[This move] does not change the image of this element in the quotient group $Q=\ast_\alpha\, \pi_1(A_\alpha)/N$, by the definition of $N$
This is the step at which Hatcher is using the hypothesis that $N$ is normal. In particular, if $N$ were simply the subgroup generated by the elements $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega)^{-1}$ (instead of the normal subgroup generated by these elements), this move would not necessarily preserve the image of this element in $G/N$.
I do not follow why this move does not change the image of the element in the quotient group $Q$. As I understand it, we have some word in $\ast_\alpha\pi_1(A_\alpha)/N$, (say) $[f_1][f_2]\cdots[f_k]$, and observe that one of the letters lies in the intersection of two of the groups in the free product, i.e. $[f_i]\in\pi_1(A_\alpha\cap A_\beta)$.
This means that $i_{\alpha\beta}(f_i)i_{\beta\alpha}(f_i)^{-1}$ is one of the generators of $N$. But why does it follow that changing the representative of $[f_i]$ in the word $[f_1][f_2]\cdots[f_k]$ does not affect the coset $N[g]$ in which $[f_1][f_2]\cdots[f_k]$ lies? In reply to the answer quoted above, what can go wrong if $N$ is not normal?
$N$ is normal if and only if $G/N$ is a group.
In any group $G$ with normal subgroup $N$, if $ab^{-1} \in N$, then $\overline{xay} = \overline{xby}$ in $G/N$ for any $x, y \in G$. This follows by cancelling the $\overline{x}$ and the $\overline{y}$ and then noting that $\overline{n} = e$ in $G/N$ for any $n \in N$, so $ab^{-1} \in N$ implies $\overline{a} = \overline{b}$ in $G/N$.