\begin{definition}[Definition:Normalizer]
Let $G$ be a group.
Let $S$ be a subset of $G$.
Then the '''normalizer of $S$ in $G$''' is the set $\map {N_G} S$ defined as:
:$\map {N_G} S := \set {a \in G: S^a = S}$
where $S^a$ is the $G$-conjugate of $S$ by $a$.
If $S$ is a singleton such that $S = \set s$, we may also write $\map {N_G} s$ for $\map {N_G} S = \map {N_G} {\set s}$, as long as there is no possibility of confusion.
\end{definition}