Show H is normal to G iff $N_G(H)=G$

Question

Let $G$ be a group and $H\leq G$. The normalizer of $H$ in $G$ is $N_G(H)=\{ g\in G |gHg^{-1} =H \}$. Show H is normal to G iff $N_G(H)=G$

I know that $H$ is normal to $N_G (H)$, for the inverse direction, if $N_G (H)=G$, then $H$ is normal to $G$.

Attempt: Since $N_G (H)=G$, and $H$ is normal to $N_G (H)$ by construction, thus $H$ is normal to $N_G (H)$

For the foreword direction, if $ H$ is normal to $G$, then $N_G(H)=G$

Attempt: Suppose $H$ is normal to $G$, then $gHg^{-1}=H, \forall g \in G$, and since $H$ is normal to $N_G (H)$, we know that $H = gHg^{-1}, \forall g \in G$, hence, $N_G(H)=G$

I don't think my proof is valid, can anyone show me how to do it?

Thanks.

Answer 1

From the definition of $N_G(H)$, we see that if $N_G(H) = G$, that $H \lhd G$, that direction is fairly trivial.

Your proof in the other direction is dangerously close to being circular reasoning. What we CAN say is that if $H \lhd G$, then $G \subseteq N_G(H)$. However, since $N_G(H)$ is also a subset of $G$, the two sets must then be equal.

Answer 2

The statement follows directly from the definition.

If $N_G(H)=G$, then for every $g\in G$ we have $g^{-1}Hg=H$, hence $H$ is normal.

Now if $H$ is normal, then for every $g\in G$ we have $g^{-1}Hg=H$, therefore $N_G(H)=G$. $\square$