Exercise 21, § 7

Let $H$ be a subgroup of finite index of a group $G$. Show that $H$ is of finite presentation if and only if the same is true of $G$. (One may suppose $G$ of finite type, cf. I, p. 125, exerc. 14. Write $G$ in the form $G = F/R$, with $F$ free of finite type; the inverse image $F'$ of $H$ in $F$ is free of finite type, cf. exerc. 20, and one has $H = F'/R$. 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 $yr_jy^{-1}$ for $y \in Y, j \in J$, where $Y$ denotes a system of representatives in $F$ of the right classes modulo $F'$.)