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.

$(1)$ iff $(5)$
See Group is Abelian iff Opposite Group is Itself.

Hence all $5$ statements are logically equivalent.