Theorem. Let $H$ be a normal subgroup of $G$. Then $\gamma:G\rightarrow G/H$ given by $\gamma(x)=xH$ is a homorphism with kernel $H$.
My question is in proving that $H$ is indeed the kernel of $\gamma$. It says:
Since $xH=H$, if and only if $x\in H$, we see that the kernel of $\gamma$ is indeed $H$.
I undertstand the statement for $xH=H$, but why does this prove that $H$ is the kernel? Why does this say that $\gamma[H]=e'$, where $e'$ is the identity in the factor group $G/H$?