Group theory MOC

Normalizer in a group

Let 𝐺 be a group and 𝑆 ⊆𝐺 be a subset. An element 𝑔 ∈𝐺 normalizes 𝑆 iff it leaves 𝑆 invariant under conjugation, i.e.

𝑔𝑆𝑔−1=𝑆

The normalizer N𝐺⁡(𝑆) of 𝑆 in 𝐺 is the subgroup of all elements normalizing 𝑆, group i.e.

N𝐺⁡(𝑆)={𝑆∈𝑃:𝑔𝑆𝑔−1=𝑆}

Proof of subgroup

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

See also

• Normalizer in a Lie algebra

---

tidy | en | SemBr