Normalizer of Conjugate is Conjugate of Normalizer

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

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

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:

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