Exercise 16, § 7

A group is called finitely generated (resp. finitely presented) if it admits a presentation $\langle (t_i)_{i \in I}, (r_j)_{j \in J} \rangle$ where I is a finite set (resp. where I and J are finite sets).

(a) Let G be a finitely presented group and $x = (x_\alpha)_{\alpha \in A}$ a finite generating family of G. Show that there exists a finite family $s = (s_k)_{k \in K}$ of relators of the family x such that $(x, s)$ is a presentation of G. (Use the preceding exercise.)

(b) Show that a finite group is finitely presented.

(c) Give an example of a finitely presented group with a subgroup which is not finitely generated.

(d) If G is a finitely presented group and H a normal subgroup of G, show the equivalence of the following properties:

(i) G/H is finitely presented.

(ii) There exists a finite subset X of G such that H is the normal subgroup of G generated by X.

(iii) If $(H_i)_{i \in I}$ is a right directed family of normal subgroup of G whose union is equal to H, there exists $i \in I$ such that $H_i = H$.

(Use (a) to prove that (i) implies (ii).)

(e) If G is a finitely presented group, show that $G/C^r(G)$ is for all $r \geq 1$. (Use § 6, Exercise 15.)

(f) Let $(G_\alpha)_{\alpha \in A}$ be a family of groups and let $G = \prod_{\alpha \in A} G_\alpha$ be the product of the $G_\alpha$. Show that G is finitely generated (resp. finitely presented) if and only if each of the $G_\alpha$ is finitely generated (resp. finitely presented) and $G_\alpha = \{e\}$ for all but a finite number of $\alpha$.