Group homomorphism/Kernel/Normal subgroup/Fact/Proof

Proof

By fact, we know that the kernel is a subgroup. We use fact. Hence, let ${}x\in G$ be arbitrary, and ${}h\in \operatorname {kern} \varphi$. Then

${}\varphi {\left(xhx^{-1}\right)}=\varphi (x)\varphi (h)\varphi {\left(x^{-1}\right)}=\varphi (x)e_{H}\varphi {\left(x^{-1}\right)}=\varphi (x)\varphi (x)^{-1}=e_{H}\,,$

therefore, ${}xhx^{-1}$ belongs to the kernel.

To fact