Equivalent Characterizations of Abelian Group

Theorem
Let $G$ be a group.

The following statements are equivalent:
 * $G$ is Abelian
 * $\forall a, b \in G: \left({a b}\right)^{-1} = a^{-1} b^{-1}$
 * $\forall a, b, c \in G: a b = c a \implies b = c$ (the Cross Cancellation Property)
 * $\forall a, b, c, d, x \in G: a x b = c x d \implies a b = c d$ (the Middle Cancellation Property).

Proof

 * Suppose that $\forall a, b \in G: \left({a b}\right)^{-1} = a^{-1} b^{-1}$.

Then:

Thus $G$ is abelian.


 * Conversely, suppose $G$ is abelian and $a, b \in G$.

By the Socks-Shoes Property, $\left({a b}\right)^{-1} = b^{-1} a^{-1}$.

Since $G$ is abelian, $b^{-1} a^{-1} = a^{-1} b^{-1}$.

Thus, $\left({a b}\right)^{-1} = a^{-1} b^{-1}$.


 * Suppose that $\forall a, b, c \in G: a b = c a \implies b = c$ (this is called the Cross Cancellation Property).

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, $b = c$.


 * Suppose that $\forall a, b, c, d, x \in G: a x b = c x d \implies a b = c d$ (this is called the middle cancellation property).

Then:

Thus $G$ is abelian.


 * Conversely, suppose $G$ is abelian and $a, b, c, d, x \in G$.

Then:

Thus the Middle Cancellation Property holds in $G$.