Exercise 29, § 7

Let $G = G_1 *_A G_2$ be the sum of two groups $G_1$ and $G_2$ amalgamated by a subgroup $A$.

(a) Show that $G$ is finitely generated if $G_1$ and $G_2$ are finitely generated.

(b) Suppose $G_1$ and $G_2$ are finitely presented. Show the equivalence of the two following properties: (1) $G$ is finitely presented. (2) $A$ is finitely generated.

(Show first that (1) implies (2). On the other hand, if $A$ is not finitely generated, there exists a right directed family $(A_i)_{i \in I}$ of subgroups of $A$ with $\bigcup_{i \in I} A_i = A$ and $A_i \neq A$ for all $i$. Let $H_i$ (resp. $H$) be the kernel of the canonical homomorphism of $G_1 * G_2$ onto $G_1 *_{A_i} G_2$ (resp. onto $G$). Show that $H$ is the union of the $H_i$ and that $H_i \neq H$ for all $i$; then apply Exercise 16(d).)

(c) Deduce an example of a finitely generated group which is not finitely presented. (Take $G_1$ and $G_2$ to be free groups of rank 2 and $A$ a free group of finite rank, cf. Exercise 22.)