Exercise 27, § 7

Let $G = *_A G_i$ be the sum of a family $(G_i)_{i \in I}$ of groups amalgamated by a common subgroup A. For all $i \in I$, choose a subset $P_i$ of $G_i$ satisfying condition (A) of no. 3. (a) Let $x \in G$ and let $(a; i_1, \ldots, i_n; p_1, \ldots, p_n)$ be a reduced decomposition of x; then

Show that, if $i_1 \neq i_n$, the order of $x$ is infinite. Show that, if $i_1 = i_n$, $x$ is conjugate to an element with a decomposition of length $\leq n - 1$.

(b) Deduce that every element of $G$ of finite order is conjugate to an element of one of the $G_i$. In particular, if the $G_i$ have no element of finite order except $e$, the same is true of $G$.