what does it mean that a subgroup self-normalizes?

Question

I am having difficulty understanding what it means for a subgroup to self-normalize. That is, given $G$ a group, and $H$ a subgroup, $N_G(H)=H$.

I've always taken $N_G(H)$ to be a subgroup of $G$ that properly contains $H$, and if $N_G(H)=G$, then $H$ is normal in $G$. So what does it mean that $H$ self-normalizes?

Answer 1

One way to look at it: the normalizer of $H$ can be defined as the largest subgroup of $G$ in which $H$ becomes normal. So if it is self-normalizing then beyond $H$ there are no elements in $G$ that normalize $H$.