Definition:Normalizer

Definition
Let $G$ be a group.

Let $S$ be a subset of $G$.

Then the normalizer of $S$ in $G$ is the set $N_G \left({S}\right)$ such that:


 * $N_G \left({S}\right) = \left\{{a \in G: S^a = S}\right\}$

where $S^a$ is the $G$-conjugate of $S$ by $a$.

If $S$ is a singleton such that $S = \left\{{s}\right\}$, we may also write $N_G \left({s}\right)$ for $N_G \left({S}\right) = N_G \left({\left\{{s}\right\}}\right)$, as long as there is no possibility of confusion.

Also denoted as
The notation $N \left({S; G}\right)$ is sometimes seen for the normalizer of $S$ in $G$.

Also see

 * Normalizer is Subgroup

Linguistic Note
The UK English spelling of normalizer is normaliser.