Exercise 16, § 7

A group is said to be of finite type (resp. of finite presentation) 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 group of finite presentation, and let $\mathbf{x} = (x_\alpha)_{\alpha \in A}$ be a finite generating family of G. Now prove that there exists a finite family $\mathbf{s} = (s_k)_{k \in K}$ of relators of the family $\mathbf{x}$ such that $(\mathbf{x}, \mathbf{s})$ is a presentation of G. (Use the preceding exercise.) b) Now prove that a finite group is of finite presentation. c) Give an example of a group of finite presentation having a subgroup which is not of finite type. d) If G is a group of finite presentation, and H a normal subgroup of G, now prove the equivalence of the following properties: (i) G/H is of finite presentation. (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 an increasing filtered family of normal subgroups 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 group of finite presentation, now prove that the same is true of $G/C^r(G)$ for all $r \geq 1$. (Use I, p. 137, exerc. 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 $G_\alpha$. Now prove that G is of finite type (resp. of finite presentation) if and only if each of the $G_\alpha$ is of finite type (resp. of finite presentation) and $G_\alpha = \{e\}$ for all $\alpha$ except a finite number of indices.