Subgroup equals Conjugate iff Normal

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $N$ be a subgroup of $G$.

Then $N$ is normal in $G$ (by definition 1) iff:
 * $\forall g \in G: g \circ N \circ g^{-1} = N$
 * $\forall g \in G: g^{-1} \circ N \circ g = N$

Also see

 * Equivalence of Definitions of Normal Subgroup