Group Abelian iff Cross Cancellation Property

Theorem
Let $G$ be a group.

Then the following are equivalent:


 * $(1): \quad G$ is abelian
 * $(2): \quad G$ has the cross cancellation property

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

$(2) \implies (1)$
Suppose that $G$ has the cross cancellation property.

Then, for all $x, y \in G$:

Thus, $G$ is abelian.

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

Let $a, b, c \in G$ be such that $a b = c a$.

Since $G$ is abelian, $c a = a c$.

We conclude that:


 * $a b = c a = a c$

Thus, by left cancellation, $b = c$.