Group homomorphism/Injectivity and kernel/Fact/Proof

Proof

If ${}\varphi$ is injective, then every element ${}h\in H$ is hit by at most one element from ${}G$. As ${}e_{G}$ is sent to ${}e_{H}$, no further element can be sent to ${}e_{H}$. Therefore, ${}\operatorname {kern} \varphi =\{e_{G}\}$. Now assume that this holds. Let ${}g,{\tilde {g}}\in G$ be elements mapping to ${}h\in H$. Then

${}\varphi {\left(g{\tilde {g}}^{-1}\right)}=\varphi (g)\varphi ({\tilde {g}})^{-1}=hh^{-1}=e_{H}\,;$

hence, ${}g{\tilde {g}}^{-1}\in \operatorname {kern} \varphi$, and so ${}g{\tilde {g}}^{-1}=e_{G}$ by the condition. Therefore, ${}g={\tilde {g}}$.

To fact