Exercise 21, § 7

Let H be a subgroup of a group G of finite index. Show that H is finitely presented if and only if G is. (It may be assumed that G is finitely generated, cf. § 4, Exercise 14. Write G in the form $G = F/R$ with F free and finitely generated; the inverse image $F'$ of H in F is free and finitely generated, cf. Exercise 20, and $H = F'/R$. If R is generated (as a normal subgroup of F) by $(r_j)_{j \in J}$, show that it is generated (as a normal subgroup of $F'$) by the $yr_jy^{-1}$, $y \in Y, j \in J$, where Y denotes a representative system in F of the right cosets modulo $F'$.)