Question about homomorphism of groups $G,H$ are groups, $\varphi:G\to H$ is homomorphism.
How do I prove: $$\ker\varphi=\left\{{e_G}\right\} \Leftrightarrow \varphi\;\text{is injective} $$
I have problem with $\Rightarrow$ direction.
Thank you.

Assume $\varphi(x)=\varphi(y)$, we are going to prove that this implies $x=y$, given that $\ker\varphi = \{e_G\}$.
Hint: What can you say about $\left(\varphi(y)\right)^{-1}=?$ Try multiplying by it on both sides.
