Normal Subgroup

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.