My question is whether or not it's true that $(\mathbb{Z} * \mathbb{Z} )/ \mathbb{Z} = (\mathbb{Z} \times \mathbb{Z})/\mathbb{Z} = \mathbb{Z}$?
The problem I'm having in my head is trying to reconcile the following intuitions:
- That it should be the case that $$(\mathbb{Z} * \mathbb{Z}) / \mathbb{Z} = \langle \, a, b \mid b=1 \rangle = \langle a \rangle = \mathbb{Z}$$
- That the map $$ \begin{align} \phi : (\mathbb{Z} \times \mathbb{Z}) / \mathbb{Z} & \to \mathbb{Z} \\ (a,b) & \mapsto a \end{align}$$ should be a homomorphism such that $\ker(\phi) = \{ (0,b) | b \in \mathbb{Z} \} = \mathbb{Z}$
- That $\mathbb{Z} * \mathbb{Z} $ is much bigger and more complex than $\mathbb{Z} \times \mathbb{Z}$ and taking the quotient of the same set should not result in the same group (I'm not sure if this is a general result)
Now, I'm not sure if I have been a bit too liberal in notation and should more concretely describe how $\mathbb{Z}$ is a subgroup in the free and direct products i.e. rewriting $(\mathbb{Z} * \mathbb{Z}) / \mathbb{Z}$ as $ \langle a, b \rangle / \langle b \rangle$.
Either way I feel like i'm missing something fundamental how we take quotient groups and what those quotient groups tell us about the original group.