Equivalent Characterizations of Abelian Group

Theorem
Let $$G$$ be a group.

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

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 $$\forall a, b, c \in G: a b = c a \Longrightarrow b = c$$ (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 $$\forall a, b, c, d, x \in G: a x b = c x d \Longrightarrow a b = c d$$ (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 $$G$$.