Normal Subgroup - Knowlpedia

Let $G$ be a group and let $H\le G$ be a subgroup. The subgroup $H$ is normal in $G$ , written $H\trianglelefteq G$ , if for every $g\in G$ ,

gHg^{-1} = H.

Normality says that $H$ is stable under the conjugation action of $G$ on itself. Equivalently, $H$ is normal iff every left coset of $H$ equals the corresponding right coset, and this is exactly the hypothesis needed to form the quotient group $G/H$ .

Examples

The center $Z(G)$ is normal in $G$ .
$A_n$ is normal in $S_n$ for all $n\ge 2$ .
Any subgroup of index $2$ is normal (see subgroup of index 2 is normal).
(Non-example) In $S_3$ , the subgroup $\{e,(12)\}$ is not normal.