Normalizer of a subset of a group

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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 $G$ .