Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 9 / a / Solution

The kernel of a group homomorphism isThe homomorphism sends identity to identity and inverses to inverses. Thus $e_G\in K$, and if $k_1,k_2\in K$, then $\Phi (k_1{k}_2^{-1})=e_{G^{\prime }}$, so $K$ is a subgroup by the subgroup test.

For any $g\in G$ and $k\in K$,Hence $gK{g}^{-1}\subseteq K$. Applying the same inclusion with $g^{-1}$ gives equality, provingThis normality argument does not require either group to be finite.