Exercise 15, § 7

Let $G$ be a group with presentation $(\mathbf{t}, \mathbf{r})$, where $\mathbf{t} = (t_i)_{i \in I}$ and $\mathbf{r} = (r_j)_{j \in J}$; let $F((T_i)_{i \in I})$ denote the free group $F(I)$ and $\mathbf{T} = (T_i)$. On the other hand, let $\mathbf{x} = (x_\alpha)_{\alpha \in A}$ be a generating family of $G$; write

$\mathbf{X} = (X_\alpha)$ and $R_\mathbf{x}$ the set of relators of $\mathbf{x}$; it is a normal subgroup of $F(A)$. For all $i \in I$, let $\phi_i(\mathbf{X})$ be an element of $F(A)$ such that $t_i = \phi_i(\mathbf{x})$ in $G$; for all $\alpha \in A$, let $\psi_\alpha(\mathbf{T})$ be an element of $F(I)$ such that $x_\alpha = \psi_\alpha(\mathbf{t})$ in $G$. Show that $R_\mathbf{x}$ is the normal subgroup of $F(A)$ generated by the elements $X_\alpha^{-1} \psi_\alpha(\phi_i(\mathbf{X}))_{i \in I}$, $\alpha \in A$ and $r_j(\phi_i(\mathbf{X}))_{i \in I} j \in J$. (If $R'_\mathbf{x}$ denotes the normal subgroup of $F(A)$ generated by the elements in question, then $R'_\mathbf{x} \subset R_\mathbf{x}$; on the other hand, show that the homomorphism $F(I) \to F(A)$ defined by the $\phi_i$ gives, when passing to the quotient, a homomorphism $G \to F(A)/R'_\mathbf{x}$ the inverse of the canonical homomorphism $F(A)/R'_\mathbf{x} \to F(A)/R_\mathbf{x} = G.$)