Equivalent Characterizations of Abelian Group

Theorem
Let $G$ be a group.

1 iff 2
See Inversion Mapping is Automorphism iff Group is Abelian.

1 iff 3
See Group is Abelian iff it has Cross Cancellation Property.

1 iff 4
See Group is Abelian iff it has Middle Cancellation Property.

Hence all four statements are logically equivalent.