Exercise 17, § 7

a) Prove that every subgroup of a finitely generated nilpotent group is finitely generated.

(In the commutative case, reason 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) Prove 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 ⊃ Cr(F(X)); use a) to show that R/Cr(F(X)) is finitely generated; observe that F(X)/Cr(F(X)) is finitely presented, cf. exerc. 16.)