Group is Abelian iff it has Middle Cancellation Property

Theorem
Let $G$ be a group.

Then $G$ is abelian iff $\forall a, b, c, d, x \in G: a x b = c x d \implies a b = c d$.

This is known as the middle cancellation property.

Proof

 * Suppose that $\forall a, b, c, d, x \in G: a x b = c x d \implies a b = c d$.

Then:

Thus $G$ is abelian.


 * Conversely, suppose $G$ is abelian and $a, b, c, d, x \in G$.

Then:

Thus the Middle Cancellation Property holds in $G$.