If $G$ is a group and $H_1\subset H_2$, where $H_1, H_2$ are subgroups of $G$, then what is the relation between $N(H_1)$ and $N(H_2)$?
Note that $N(H_1)=\{x\in G : xH_1=H_1x\}$ is the $\textbf{normalizer}$ of $H_1$ and similarly, $N(H_2)$ is the $\textbf{normalizer}$ of $H_2$.
Here's my answer: $x\in N(H_1) \Rightarrow xH_1=H_1x \Rightarrow xH_2=H_2x \Rightarrow x\in N(H_2)$ which implies $N(H_1)\subset N(H_2)$.
However, I am not sure if my logic is correct. Something tells me it is otherway, ie $N(H_2)\subset N(H_1)$.
Please confirm whether this is correct.
Best Answer
Given that the normalizer of $\{e\}$ is $G$ and the normalizer of $G$ is also $G$ any result in that vein would give us the normalizer of any subgroup is also $G$.