Normal Subgroup

normal

Let $H \leq G$ be normal if:

N_{G} (H) = G

(see Normalizer). Which means you get the equivalence of Equivalences for Normal Subgroups.

We'll use the notation $K ⊴ G$ when $K$ is normal to $G$ .

Example

Recall our Left and Right Cosets#Examples $G = S_{3}$ with $H = ⟨ (1 2) ⟩ = {1, (1 2)}$ . Then we saw that:

(1 3) H \neq H (1 3)

thus $H$ is not normal. On the other hand if we have:

K = ⟨ (1 2 3) ⟩ = {1, (1 2 3), (1 3 2)}

is normal.