Exercise 17, § 7

(a) Show that every subgroup of a finitely generated nilpotent group is finitely generated. (In the commutative case, argue by induction on the minimum number of generators of the group; then proceed by induction on the nilpotency class of the group.)

Give an example of a finitely generated solvable group containing a subgroup which is not finitely generated.

(b) Show that every finitely generated nilpotent group is finitely presented. (Write the group in the form F(X)/R, with X finite, and choose r such that $R \supset C^r(F(X))$; use (a) to show that $R/C^r(F(X))$ is finitely generated; observe that $F(X)/C^r(F(X))$ is finitely presented, cf. Exercise 16.)