Group is Abelian iff it has 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 $x, y \in G$:

\(\ds y \paren {x y}\)

\(=\)

\(\ds \paren {y x} y\)

Group Axiom $\text G 1$: Associativity

\(\ds \leadsto \ \ \)

\(\ds x y\)

\(=\)

\(\ds y x\)

Definition of Cross Cancellation Property

Thus, $G$ is abelian.

$\Box$

$(1) \implies (2)$

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, $b = c$.

$\blacksquare$