Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 3 / 5E / i / 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 $g\in H$. Thereforewhich proves the implication from normality to being the kernel of a group homomorphism.