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