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$\begingroup$

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?

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    A free group does not have any non-identity elements whihc are of finite order.2012-08-21
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    Edit note: it looks like the original post had $\ast$ and then an editor changed them all to $\times$, and @Benjalim restored one to $\ast$. I just restored the other one to $\ast$ as well.2012-08-21
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    It could help, perhaps, to know what definition of free group you have.2012-08-21
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    @Belgi I suggest if you don't know what a free group is to ***not*** change the symbols.2012-08-21

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