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 $G$ (by definition 1) iff:


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

or equivalently:
 * The right coset space of $N$ in $G$ equals its left coset space.

Proof
If $N$ is a normal subgroup of $G$ by Definition 1 then the equality of the coset spaces follows directly from definition of normal subgroup and coset.

Suppose that every right coset of $N$ in $G$ is a left coset of $N$ in $G$.

Let $g \in G$

Since every right coset of $N$ in $G$ is a left coset, there exists an $h \in G$ such that $N \circ g = h\circ N$.

Then since $e \in N$, $g \in h \circ N$.

Thus for some $n \in N$, $g = h \circ n$, so $g^{-1} = n^{-1} \circ h^{-1}$, so $g^{-1} \circ h = n^{-1}$, so $g^{-1} \circ h \in N$.

Then

Also see

 * Equivalence of Definitions of Normal Subgroup