I have the following two groups ,
$G=\mathbb{Z}*_\mathbb{Z} \mathbb{Z}$, that is free product of two copies of $\mathbb{Z}$ amalgamated over $\mathbb{Z}$.
$H$ is the group which is given by the following presentation $$H=<a,b\;|\;abab^{-1}>$$
First of all I want to know whether these two groups are isomorphic or not. If yes, how should I prove it?
As the question is currently written, there is not a unique answer to it.
The structure of $G*_{F}H$ depends on the two homomorphisms $\phi\colon F\rightarrow G$ and $\psi\colon F\rightarrow H$ given, as one has: $$G*_FH=(G*F)/N,$$ where $N$ is the normal subgroup of $G*H$ generated by $\{\phi(f)\psi^{-1}f;f\in F\}$.
Therefore, $\mathbb{Z}*_{\mathbb{Z}}\mathbb{Z}$ is ambiguous as they are infinitely homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}$.