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 of one of the $G_i$, and let $\Theta$ be the union of the elements of $S$. (a) Let $H, H'$ be in $S$, such that $H \neq H'$, and let $x \in H - \{e\}$, $x' \in H' - \{e\}$. Now prove that $xx' \in G - \Theta$. (Reduce to the case where $H$ is one of the $G_i$; write $H'$ in the form $yG_jy^{-1}$ and proceed as in the preceding exercise.) (b) Let $C$ be a subgroup of $G$ contained in $\Theta$. Now prove 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. Now prove that $C$ is contained in $\Theta$ (cf. I, p. 148, exerc. 27); deduce that $C$ is contained in a conjugate of a $G_i$.