Normalizer in a group

Let $𝐺$ be a group and $𝑆 \subseteq 𝐺$ be a subset. An element $𝑔 \in 𝐺$ normalizes $𝑆$ iff it leaves $𝑆$ invariant under conjugation, i.e.

𝑔 𝑆 𝑔 - 1 = 𝑆

The normalizer $N 𝐺 (𝑆)$ of $𝑆$ in $𝐺$ is the subgroup of all elements normalizing $𝑆$ , #m/def/group i.e.

N 𝐺 (𝑆) = {𝑆 \in 𝑃 : 𝑔 𝑆 𝑔 - 1 = 𝑆}

Proof of subgroup

This is just the setwise stabilizer of $𝑆$ under the conjugation action.

See also