Group is Abelian iff it has Cross Cancellation Property

Theorem
Let $G$ be a group.

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

This is known as the cross cancellation property.

Proof

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

Then:

Thus, $G$ is abelian.


 * Conversely, suppose $G$ is abelian.

Let $a, b, c \in G$ where $a b = c a$.

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

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