Description of Homomorphisms

Let $G$ be a fininte Group of order $n$ for which we have a presentation and let $S=\{s_1,\dots ,s_m\}$ be the generators. Let $H$ be another group and $\{r_1,\dots ,r_m\}$ be elements of $H$. Suppose that any relation satisfied in $G$ by the $s_i$ is also satisfied in $H$ where each $s_i$ is replaced by $r_i$. Then there is a unique homomorphism $\phi :G\to H$ which maps $s_i$ to $r_i$. If we have a presentation for $G$ then we need only check the relations specified by this presentation. Here:

• $\phi$ is surjective if when $H$ is generated by all $\{r_1,\dots ,r_{mm}\}$
• $\phi$ is further injective when $|G|=|H|$, thus an Isomorphism.