Prove that G φ H is one-to-one if and only if Ker(φ) consists only of e, and onto if and only if

Question

Prove that $G \xrightarrow{\varphi} H$ is one-to-one if and only if $\operatorname{Ker}(\varphi)$ consists only of $e$, and onto if and only if $\operatorname{Im}(\varphi)=H$.

   Prove that $G \xrightarrow{\varphi} H$ is one-to-one if and only if $\operatorname{Ker}(\varphi)$ consists only of $e$, and onto if and only if $\operatorname{Im}(\varphi)=H$.

Mathematical physics

Robert Geroch 1st Edition

Chapter 6, Problem 46

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Prove that $G \xrightarrow{\varphi} H$ is one-to-one if and only if $\operatorname{Ker}(\varphi)$ consists only of $e$, and onto if and only if $\operatorname{Im}(\varphi)=H$.