Some basic terms from finite group theory, normalising and centralising

Question

In a proof a read the following "Since $H$ and $N$ normalize each other, if follows that $H \subseteq C_G(N)$". I thought normalising just means that one subgroup is normal with respect to the other, but that does not seems to be meant here.

So what does it mean to say that a subgroup $A$ normalises another subgroup $B$?

And further, it is often read that a subgroup $A$ centralises another subgroup $B$, and I guess this means that $A \le C_G(B)$, right?

Answer 1

To say "$H$ normalizes $N$" means that for each $h \in H$ we have $hNh^{-1}=N$. Neither $H$ nor $N$ needs to be a subgroup of the other for this to make sense, although both have to be a subgroup of the same group $G$.

Answer 2

Let $H,N$ normalize each other and $H\cap N =e$ then $H \subseteq C_G(N)$;

Let $x=hnh^{-1}n^{-1}$ since $H$ normalize $N$, $hnh^{-1}\in N \implies x\in N$

since $N$ normalize $H$, $nh^{-1}n^{-1}\in H \implies x\in H$

As a result $x\in H\cap N \implies x=e \implies hn=nh$.