Normalizer $N_G(S)$ of a subset

Let \(G\) be a group and \(S\subseteq G\) a subset, and for \(g\in G\) write \(gSg^{-1}\coloneqq\theset{gsg^{-1} \st s\in S}\). The normalizer of \(S\) in \(G\) is \[ N_G(S) \coloneqq \theset{g\in G \st gSg^{-1} = S}. \]