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.

Facts