Group is Abelian iff it has Middle Cancellation Property

Theorem
Let $$G$$ be a group.

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

This is known as the middle cancellation property.

Proof

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

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 $$G$$.