Equivalent Characterizations of Abelian Group

Theorem
Let $$G$$ be a group. The following statements are equivalent $$
 * $$G$$ is Abelian;
 * $$\forall a,b \in G,$$ $$(ab)^{-1}=a^{-1}b^{-1}$$;
 * (Cross Cancellation) $$\forall a,b,c \in G,$$ $$ab=ca \Longrightarrow b=c$$;
 * (Middle Cancellation) $$\forall a,b,c,d,x \in G,$$ $$axb=cxd \Longrightarrow ab=cd.

Proof
Suppose that $$\forall a,b \in G,$$ $$(ab)^{-1}=a^{-1}b^{-1}$$. Then,

Thus, $$G$$ is Abelian.

Conversely, suppose $$G$$ is Abelian and $$a,b \in G$$. By the Socks-Shoes Property, $$(ab)^{-1}=b^{-1}a^{-1}$$, and since $$G$$ is Abelian, $$b^{-1}a^{-1}=a^{-1}b^{-1}$$. Thus, $$(ab)^{-1}=a^{-1}b^{-1}$$.

Suppose that $$\forall a,b,c \in G,$$ $$ab=ca \Longrightarrow b=c$$ (this is called the cross cancellation property). Then,

Thus, $$G$$ is Abelian.

Conversely, suppose $$G$$ is Abelian. Let $$a,b,c \in G$$ where $$ab=ca$$. Since $$G$$ is Abelian, $$ca=ac$$, so $$ab=ca=ac$$. Thus, by left cancellation, $$b=c$$.

Suppose that $$\forall a,b,c,d,x \in G,$$ $$axb=cxd \Longrightarrow ab=cd $$ (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, middle cancellation holds in $$G$$.