Understand significance of theorem related to normal subgroups I found in an online book about abstract algebra the following theorem:

The following theorem is fundamental to our understanding of normal subgroups.
Theorem 10.3.  Let $G$ be a group and $N$ be a subgroup of $G$. Then the following statements are equivalent.

*

*The subgroup $N$ is normal in $G$.

*For all $g \in G\text{,}$ $gNg^{-1} \subset N\text{.}$

*For all $g \in G\text{,}$ $gNg^{-1} = N\text{.}$

Since I'm learning right now about normal groups and factor groups, I wonder why the author says that the "theorem is fundamental to understanding normal subgroups."
I found the proofs rather complicated, because we first show that $gNg^{-1} \subset N$, then the other way round that $N \subset gNg^{-1}$ to conclude $gNg^{-1} = N$ for a subgroup $N$ that is normal in a group $G$.
Somehow I fail to understand why this is so fundamental, can someone explain it?

The map $G \to G$, $h \to ghg^{-1}$ is important in group theory, and is known as conjugation by $g$. The significance of the theorem is in showing normal subgroups are exactly the subgroups of $G$ which are invariant under conjugation by any element $g \in G$.
