A question about the proof of Seifert - van Kampen

I'm studying Algebraic Topology using Hatcher's textbook as my main reference, and there is a detail in the proof of the Seifert - van Kampen Theorem (page 43) which is still unclear to me. The heart of the proof of the second part of the statement (the one regarding relations) is where it says (page 45):

If we can show that any two factorizations of $[f]$ are equivalent, this will say that the map $Q \rightarrow \pi_1(X)$ induced by $\phi$ is injective, hence the kernel of $\phi$ is exactly $N$, and the proof will be complete.

I'm not exactly sure which result Hatcher is referring to here, but the way I see it, this statement is a sort of a converse of the classical homomorphism theorem (if $\phi: G \rightarrow G'$ is a group homomorphism then $ \phi $ canonically defines an injective map ${G}/{\ker \phi} \rightarrow G'$). To make things rigorous I stated the following

Lemma Let $\phi: G \rightarrow G'$ be a group homomorphism and $N \leq \ker \phi $ ($N$ not necessarily normal in $G$). Let $G/N^r$ be the set of right cosets of $N$. Then we can define \begin{align} \tilde{\phi}: G/N^r &\rightarrow G' \\ Nx &\mapsto \phi(x) \end{align} and if $\tilde{\phi}$ is injective then $\ker \phi = N$.

I managed to prove this lemma (if necessary I will edit the question to include such proof), but the problem now is that this seems to be too strong a result! What the Theorem says is that the kernel of $\Phi$ is the normal subgroup generated by all the $i_{\alpha \beta}(\omega) i_{\beta \alpha}(\omega)^{-1}$'s; it doesn't say that the kernel is the subgroup generated by all such elements, which also happens to be normal. By applying the above lemma to Hatcher's proof, with $G = \ast_{\alpha} \pi_1({A_\alpha)}, N = \langle \{ i_{\alpha \beta}(\omega) i_{\beta \alpha }(\omega)^{-1}: \omega \in \pi_1 (A_{\alpha} \cap A_{\beta}) \}\rangle, \phi = \Phi$ (note that $N \leq \ker \phi$ is remarked at the beginning of page 43), and by subsituting every instance of $Q = \ast_{\alpha} \pi_1(A_{\alpha})$ with the set of right cosets, one would thus be able to prove a stronger version of the theorem (taking $\ker \Phi$ to be the subgroup generated by the $i_{\alpha \beta}(\omega) i_{\beta \alpha }(\omega)^{-1}$'s , and not its normal closure; the normality of such a subgroup would then be a consequence of the fact that it is the kernel of a group homomorphism). This makes me suspicious that I might be getting something wrong...
So, to summarize, here are my questions:

• What am I getting wrong? Have I misunderstood the statement of the theorem, is the lemma wrong or is the normality of the subgroup used in some other part of the proof? Or is everything right?

• Can someone provide an explicit example in which the "stronger version" of the theorem doesn't hold: i.e. in which $ \langle \{ i_{\alpha \beta}(\omega) i_{\beta \alpha }(\omega)^{-1}: \omega \in \pi_1 (A_{\alpha} \cap A_{\beta}) \}\rangle $ is different from $\langle \{ i_{\alpha \beta}(\omega) i_{\beta \alpha }(\omega)^{-1}: \omega \in \pi_1 (A_{\alpha} \cap A_{\beta}) \}^{\ast_{\alpha} \pi_1(A_{\alpha})} \rangle $.

Thank you very much for any help.

I think the flaw in your reasoning comes earlier in the proof. 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 think it should be pointed out that another style of proof is available, and is used to give a proof of the version using many base points, at this groupoids. With the result for many base points, you can compute the fundamental group of the circle, which is, after all, THE basic example in algebraic topology. The proof given there does only the union of 2 open sets, but it gives the proof by

verification of the universal property,

which is a general procedure of great use in mathematics. For example this method is used to prove higher dimensional versions of the van Kampen Theorem. This method also avoids description of the result by generators and relations.

Ok, I can at least show why if $N$ is a normal subgroup of $*_{\alpha}\pi_{1}(A_{\alpha})$, $N\subseteq \ker\Phi \subseteq *_{\alpha}(A_{\alpha})$, and $\Phi:*_{\alpha}\pi_{1}(A_{\alpha})\rightarrow\pi_{1}(X)$ is injective, then $\ker\Phi=N$.

The reason is as follows: we have the following diagram

$$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} *_{\alpha}\pi_{1}(A_{\alpha})/N & \ra{\Phi} & \pi_{1}(X) \\ \da{f} & & \da{id} \\ *_{\alpha}\pi_{1}(A_{\alpha})/\ker\Phi & \ras{\Phi} & \pi_{1}(X) & \\ \end{array} $$

and the mapping $f$ that makes the diagram commutes is the following:

$$ \begin{eqnarray} f:&*_{\alpha}\pi_{1}(A_{\alpha})/N\rightarrow *_{\alpha}\pi_{1}(A_{\alpha})/\ker\Phi\\ & gN\rightarrow g\ker\Phi \end{eqnarray} $$

By the fact that $\Phi:*_{\alpha}\pi_{1}(A_{\alpha})/N\rightarrow \pi_{1}(X)$ is injective, this makes $f$ an injective homomorphism. On the other hand, $f$ is also surjective. By a theorem in group theory (3rd isomorphism theorem), $\ker f=(\ker\Phi)/N$. But $f$ is also injective, so $(\ker\Phi)/N=\{N\}$. Hence if $x\in\ker\Phi$, then $xN=N$ means that $x\in N$. So $\ker\Phi=N$.

I don't understand is why

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 $A_{\alpha}\cap A_{\beta}$.

would imply that

[This move] does not change the image of the element in $Q$...