Subgroup is Normal iff Left Cosets are Right Cosets

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

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

Then $N$ is normal in $G$ (by definition 1) :


 * 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.

Necessary Condition
Let $N$ be a normal subgroup of $G$ by Definition 1.

Then the equality of the coset spaces follows directly from definition of normal subgroup and coset.

Sufficient Condition
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 $N \circ g = h \circ N$.

By Element of Group is in its own Coset:
 * $g \in N \circ g = h \circ N$

From Element in Left Coset iff Product with Inverse in Subgroup:
 * $g^{-1} \circ h \in N$

Then:

Since this holds for all $g \in G$, $N$ is normal in $G$ (by definition 1).

Also see

 * Equivalence of Definitions of Normal Subgroup