I recently start reading Hatcher's book for self-study. On page $46$ it gives such an example that is a free product and not a free group.
I don't quite understand the explanation given in the book. Should I show that any subset of $\mathbb{Z}_2\ast\mathbb{Z}_2$ cannot be the basis for a free group?
Secondly, what is the relationship between free abelian groups and free groups?