Group is Abelian iff it has Middle Cancellation Property

Theorem
Let $G$ be a group.

Then the following are equivalent:


 * $(1): \quad G$ is abelian
 * $(2): \quad G$ satisfies the middle cancellation property

Proof
Let us suppress the operation of $G$ for brevity.

$(2) \implies (1)$
Suppose that $G$ satisfies the middle cancellation property.

Then, for all $g, h \in G$:

Thus $G$ is abelian.

$(1) \implies (2)$
Conversely, suppose $G$ is abelian.

Then, for all $a, b, c, d, x \in G$:

Thus the middle cancellation property holds in $G$.