Definition:Normalizer

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.

The UK English spelling of this is "normaliser".