I am studying Van Kampen Theorem using Hatcher's textbook. I am dealing with the general statement, I mean: (pg 43)
He defines previously the free product of groups (pg 41) as:
I can follow the main idea of the proof but I don't understand how he can say (pg 45):
By definition, elements of the free product should be reduced words, am I right? Then he should not be considering unreduced words in a free product, since there won't be none. I suppose I have missed something, if that is the case and I am pretty sure that it is, Where I am misunderstanding?
Thanks in advance!
The author might need to consider unreduced words in order to simplify the presentation of the proof. So while, technically speaking, it is slight abuse of language to say that these unreduced words live in the free product, one may extend the map $\Phi$ to these words by reducing them first and then applying the original $\Phi$.