Exercise 33, § 7

Let $E = G *_A G'$ be the sum of two groups amalgamated by a common subgroup A. Let P (resp. P') be a subset of G (resp. G') satisfying condition (A) of no. 3.

Let $n$ be an integer $\geq 1$. Let $L_n$ denote the set of $x \in E$ whose reduced decomposition is of length $\leq 2n - 1$. Let $M_n$ denote the set of $x \in E$ of the form $ap_1p'_1p_2p'_2 \ldots p_np'_n$, with $a \in A, p_i \in P - \{e\}, p'_i \in P' - \{e\}$; similarly, let $M'_n$ denote the set of elements of the form $ap'_1p_1p'_2p_2 \ldots p'_np_n$, where $a, p_i, p'_i$ satisfy the same conditions.

(a) Show that $L_n, M_n$ and $M'_n$ are disjoint and that their union is the set of elements of $E$ whose reduced decomposition is of length $\leq 2n$.

(b) Construct a bijection $\varepsilon$ of $M_n$ onto $M'_n$ such that $\varepsilon(ax) = a\varepsilon(x)$ if $a \in A, x \in M_n$.

(c) Let $X_n = L_n \cup M_n$ and $X'_n = L_n \cup M'_n$; show that $G.X_n = X_n$ and $G'.X'_n = X'_n$.

(d) $\varepsilon$ extends to a bijection of $X_n$ onto $X'_n$ by setting $\varepsilon(x) = x$ if $x \in L_n$. Show that $E$ can operate on $X_n$ so that $g(x) = gx$ if $g \in G, x \in X_n$ and $g'(x) = \varepsilon^{-1}(g' \in (x))$ if $g' \in G', x \in X_n$. Show that, if $g \in X_n$, then $g(e) = g$.

(e) Let $\tau_n : E \to \mathscr{G}_{X_n}$ be the homomorphism defined by the action of $E$ on $X_n$ described above and let $R_n$ be the kernel of $\tau_n$. Show that $R_n \cap X_n = \{e\}$; deduce that the intersection of the $R_n$ reduces to $e$.

(f) When $G$ and $G'$ are finite, show that $E$ is residually finite (cf. § 5, Exercise 5). (Note that the $R_n$ are then subgroups of finite index.)