Group is Abelian iff Opposite Group is Itself

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

Let $\left({G, *}\right)$ be the opposite group to $({G, \circ})$.

$\left({G, \circ}\right)$ is an Abelian group :
 * $\left({G, \circ}\right) = \left({G, *}\right)$

Proof
By definition of opposite group:


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

Necessary Condition
Let $\left({G, \circ}\right)$ be Abelian.

Then:

Sufficient Condition
Let $\left({G, \circ}\right) = \left({G, *}\right)$.

Then:

Thus by definition $\left({G, \circ}\right)$ is an Abelian group.