Subgroup is Normal iff Left Cosets are Right Cosets

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 $G$ iff:
 * Every right coset of $N$ in $G$ is a left coset.

Proof
Follows directly from definition of normal subgroup and coset.

Also see

 * Normal Subgroup Equivalent Definitions

Also as definition
Some sources use this property as the definition of a normal subgroup.