Exercise 31, § 7

Let G be the free product of a family $(G_i)_{i \in I}$ of groups. Let S be the set of subgroups of G which are conjugate to one of the $G_i$ and $\Theta$ the union of the elements of S.

(a) Let $H, H' \in S$, with $H \neq H'$, and let $x \in H - \{e\}$, $x' \in H' - \{e\}$. Show that $x x' \in G - \Theta$. (Reduce it to the case where H is one of the $G_i$; write $H'$ in the form $y G_j y^{-1}$ and proceed as in the preceding exercise.)

(b) Let C be a subgroup of G contained in $\Theta$. Show that there exists $H \in S$ such that $C \subset H$.

(c) Let C be a subgroup of G all of whose elements are of finite order. Show that C is contained in $\Theta$ (cf. Exercise 27); deduce that C is contained in a conjugate of one of the $G_i$.