Given $$G=\mathbb{Z}_2*\mathbb{Z}_2=P(a,b\mid a^2,b^2)$$ among other things I wanted to show that this group is infinite, what I did is consider the words of the form $$abababa\ldots$$ they are all diferent from each other since they do not contain any of the relations described below, and also there is an infinite amount of them.
You can do the same trick for any 2 finite-presented groups, so my question is, is the free product of two arbitrary (non trivial) groups infinite always? If so, how do you show it?
I also had to calculate $\text{Ab}(G)$ and I get $$\text{Ab}(G)=\mathbb{Z}_2\times\mathbb{Z}_2$$ is it true that, for any two abelian groups $G_1,G_2$ the following holds? $$\text{Ab}(G_1*G_2)=G_1\times G_2$$