Normalizing subgroup

Definition

Let $H$ and $K$ be subgroups of a group $G$. We say that $H$ is a normalizing subgroup for $K$ if the following equivalent conditions are satisfied:

• For every $h\in H$, $hKh^{-1}=K$
• $H\leq N_{G}(K)$, viz, $H$ is contained in the normalizer of $K$