Normalizer of Conjugate is Conjugate of Normalizer

Theorem
The normalizer of a conjugate is the conjugate of the normalizer:


 * $S \subseteq G \implies N_G \left({S^a}\right) = \left({N_G \left({S}\right)}\right)^a$

Proof
From the definition of conjugate, $S^a = \left\{{y \in G: \exists x \in S: y = a x a^{-1}}\right\} = a S a^{-1}$.

From the definition of normalizer, $N_G \left({S}\right) = \left\{{x \in G: S^x = S}\right\}$.

Thus: