[Math] Prove that the centralizer subgroup is normal in the normalizer subgroup

Question

To my dear friends with gratitude.
I want to get help proving centralizer of a nonempty subset of a group is a normal subgroup in the normalizer of that set in the mentioned group.symbolically:
$C_G (S)\trianglelefteq N_G (S)$

Answer 1

Let $z \in C_G(S)$ and let $n \in N_G(S)$. We want to show that $nzn^{-1} \in C_G(S)$.

If $s \in S$, then $(nzn^{-1})s(nzn^{-1})^{-1}=nz(n^{-1}sn)z^{-1}n^{-1}$. Now $n^{-1}sn \in S$ and so $nz(n^{-1}sn)z^{-1}n^{-1}=n(n^{-1}sn)n^{-1}=s$.