Group is Abelian iff Opposite Group is Itself

Theorem
Let $\struct {G, \circ}$ be a group.

Let $\struct {G, *}$ be the opposite group to $\struct {G, \circ}$.

$\struct {G, \circ}$ is an Abelian group :
 * $\struct {G, \circ} = \struct {G, *}$

Proof
By definition of opposite group:


 * $(1): \quad \forall a, b \in G : a \circ b = b * a$

Necessary Condition
Let $\struct {G, \circ}$ be Abelian.

Then:

Sufficient Condition
Let $\struct {G, \circ} = \struct {G, *}$.

Then:

Thus by definition $\struct {G, \circ}$ is an Abelian group.