Equivalent Characterizations of Abelian Group

Theorem
Let $G$ be a group.

Then the following statements are equivalent:

Proof
That $(1)$ is equivalent to $(2)$ is shown on Mapping to Inverse is Endomorphism iff Abelian.

That $(1)$ is equivalent to $(3)$ is proved on Group Abelian iff Cross Cancellation Property.

That $(1)$ is equivalent to $(4)$ is proved on Group Abelian iff Middle Cancellation Property.

Hence all four statements are logically equivalent.