Normalizer of a subset of a group

Definition
Let $$G$$ be a group and $$S$$ be a subset of $$G$$. The normalizer (normaliser) of $$S$$ in $$G$$, denoted $$N_G(S)$$ is defined as:

$$N_G(S) := \{ g \in G \mid gSg^{-1} = S \}$$.

Equivalently, it is the isotropy of $$S$$ under the action of $$G$$ on the set of subsets of $$G$$ by conjugation.

We typically use the term normalizer for normalizer of a subgroup, i.e., where the subset we start with is a subgroup of $$G$$.

Facts

 * Group acts on set of subsets by conjugation