Group is Abelian iff it has Cross Cancellation Property

Theorem
Let $$G$$ be a group.

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

This is known as the cross cancellation property.

Proof

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

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, $$b = c$$.