Proposition 9

Let $G$ be a group, $t = (t_i)_{i \in I}$ a generating family of $G$ and $r = (r_j)_{j \in J}$ a family of relators of $t$. The following conditions are equivalent:

(a) The ordered pair $(t, r)$ is a presentation of $G$.

(b) Let $G'$ be a group and $t' = (t'_i)_{i \in I}$ a family of elements of $G'$. If $r_j(t') = e$ for all $j \in J$, there exists a homomorphism $u$ of $G$ into $G'$ such that $u(t_i) = t'_i$ for all $i \in I$.

(c) Let $\overline{G}$ be a group and $\bar{t} = (\bar{t}_i)_{i \in I}$ a generating family of $\overline{G}$ such that $r_j(\bar{t}) = e$ for all $j \in J$. Every homomorphism of $\overline{G}$ into $G$ which maps $\bar{t}_i$ to $t_i$ for all $i \in I$ is an isomorphism.

Let $f$ denote the homomorphism of $F(I)$ into $G$ which maps $i$ to $t_i$ for all $i \in I$ and $N$ the kernel of $f$.

(a) $\Rightarrow$ (b): Suppose that $(t, r)$ is a presentation of $G$ and let $t' = (t'_i)_{i \in I}$ be a family of elements of a group $G'$ with $r_j(t') = e$ for all $j \in J$. Let $f'$ be the homomorphism of $F(I)$ into $G'$ characterized by $f'(i) = t'_i$ for all $i \in I$. By hypothesis $f'(r_j) = e$ for all $j \in J$ and, as $N$ is generated by the elements $gr_jg^{-1}$ for $j \in J$ and $g \in F(I), f'(N) = \{e\}$. As the homomorphism $f \colon F(I) \to G$ is surjective with kernel $N$, there exists a homomorphism $u \colon G \to G'$ such that $f' = u \circ f$. Then $u(t_i) = u(f(i)) = f'(i) = t'_i$.

(b) $\Rightarrow$ (c): Suppose condition (b) holds. Let $t = (t_i)_{i \in I}$ be a generating family of a group $G$ such that $r_j(t) = e$ for all $j \in J$ and let $v$ be a homomorphism of $\overline{G}$ into $G$ such that $v(\bar{t}_i) = t_i$ for all $i \in I$. As the family $(t_i)_{i \in I}$ generates $G$, the homomorphism $v$ is surjective. By property (b) there exists a homomorphism $u \colon G \to \overline{G}$ such that $u(t_i) = \bar{t}_i$ for all $i \in I$. Then $u(v(\bar{t}_i)) = \bar{t}_i$ for all $i \in I$ and hence $u \circ v$ is the identity on $\overline{G}$, which proves that $v$ is injective. Hence $v$ is an isomorphism and condition (c) holds.

(c) $\Rightarrow$ (a): Suppose condition (c) holds. Let $t'_i$ be the canonical image of $i$ in $F(I, r)$ and $t' = (t'_i)_{i \in I}$; then $r_j(t') = e$ for all $j \in J$. As $r_j(t) = e$ for all $j \in J$, there exists one and only one homomorphism $v$ of $F(I, r)$ into $G$ such that $v(t_i) = t_i$ for all $i \in I$. By (c), $v$ is an isomorphism of $F(I, r)$ onto $G$ which transforms the presentation $(t', r)$ of $F(I, r)$ into a presentation $(t, r)$ of $G$.