Normal Subgroup Criterion

Normal Subgroup Criterion: Let $G$ be a group and let $N\le G$ be a subgroup . Then $N$ is a normal subgroup of $G$ if and only if for every $g\in G$ and every $n\in N$ one has $gng^{-1}\in N$.

Equivalently, $N$ is normal if and only if $gNg^{-1}=N$ for all $g\in G$ (where $gNg^{-1}=\{gng^{-1}:n\in N\}$). This says precisely that $N$ is invariant under the conjugation action of $G$ on itself.

Proof sketch: If $N$ is normal, then left and right cosets coincide, so conjugation carries $N$ to itself. Conversely, if $gNg^{-1}\subseteq N$ for all $g$, applying the same inclusion with $g^{-1}$ gives $N\subseteq gNg^{-1}$, hence equality, which implies normality.