Exercise 32, § 7

Let $A$ and $B$ be two groups, and let $G = A * B$ be their free product. Let $X$ be the set of commutators $(a, b)$, for $a \in A - \{e\}$ and $b \in B - \{e\}$, and let $S = X \cup X^{-1}$. a) Now prove that $X \cap X^{-1} = \varnothing$, and that, if $x$ is an element of $X$, there exists a unique pair $(a, b)$ in $(A - \{e\}) \times (B - \{e\})$ such that $(a, b) = x$. b) Let $n \geq 1$, and let $(s_1, \ldots, s_n)$ be a sequence of elements of $S$ such that $s_i s_{i+1} \neq e$ for $1 \leq i < n$; let $a_n \in A - \{e\}$, $b_n \in B - \{e\}$, $\varepsilon(n) = \pm 1$ be such that $s_n = (a_n, b_n)^{\varepsilon(n)}$. Let $g = s_1 \ldots s_n$. Now prove, by induction on $n$, that the reduced decomposition of $g$ is of length $\geq n + 3$, and terminates either by $a_n b_n$ if $\varepsilon(n) = 1$, or by $b_n a_n$ if $\varepsilon(n) = -1$. c) Let $R$ be the kernel of the canonical projection $A * B \to A \times B$ corresponding (I, p. 80, prop. 4) to the canonical injections $A \to A \times B$ and $B \to A \times B$. Now prove that $R$ is a free group of basic family $X$. (Use $b$ for proving that $X$ is free; if $R_X$ is the subgroup of $G$ generated by $X$, show that $R_X$ is normal; deduce that it coincides with $R$.)