Solution
= Solution
Assume $H$ is a <normal subgroup>. Take the <quotient group> $K=G/H$ and the natural projection $\theta(g)=gH$. The well-defined multiplication constructed above gives $\theta(gk)=\theta(g)\theta(k)$, so this is a <group homomorphism>. Its identity is the <coset> $H$, and $\theta(g)=H$ exactly when $g\in H$. Therefore
$$
\boxed{\ker\theta=H,}
$$
which proves the implication from normality to being the <kernel of a group homomorphism>.