Normal subgroup/Characterization/Fact

< Normal subgroup < Characterization

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

Then the following statements are equivalent.

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

Proof, Write another proof