Normalizer

Let $G$ be a group and let $H\le G$ be a subgroup . The normalizer of $H$ in $G$ is $N_G(H) \;=\; \{\,g\in G : gHg^{-1}=H\,\}.$ It is a subgroup of $G$ containing $H$.

The normalizer is the largest subgroup $K\le G$ such that $H$ is a normal subgroup of $K$ (indeed, $H\trianglelefteq N_G(H)$ by definition). Normalizers are a basic tool for controlling conjugacy and appear throughout finite group theory (e.g. in Sylow theory).

Examples:

• If $G$ is abelian, then $N_G(H)=G$ for every subgroup $H$.
• In $S_3$, if $H=\{e,(12)\}$ then $N_{S_3}(H)=H$ (so $H$ is not normal in $S_3$).
• In $S_3$, if $H=A_3$ then $N_{S_3}(H)=S_3$ (since $A_3$ is normal in $S_3$).