Free product of infinite many groups is not finitely generated?

Question

Let $\{G_i\mid i\in I\}$ an infinite family of (not trivial) groups. Is it true that the free product $\ast_{i\in I} G_i$ is not finitely generated?

I think it's true, I just need confirmation.

Answer 1

Suppose

$$\;^*_{i\in I} G_i=\langle\;a_1,\ldots,a_n\;\rangle\;$$

But in each $\;a_i\;$ appears only a finite number of elements from a finite number of groups $\;G_i\;$ and thus the same is true for the whole set $\{a_1,...,a_n\}\;$