Normal subgroup/Characterization/Fact

Let ${}G$ be a group, and let ${}H\subseteq G$ be a subgroup.

Then the following statements are equivalent.

1. ${}H$ is a normal subgroup of ${}G$.
2. We have ${}xhx^{-1}\in H$ for all ${}x\in G$ and ${}h\in H$.
3. ${}H$ is invariant under every inner automorphism of ${}G$.

Proof, Write another proof