 Independence of the Abelian group axioms

The usual associative, identity, two-sided inverse and commutativity axioms form an independent axiom set. A three-element commutative unital operation with self-inverses can fail associativity; a constant-zero operation can satisfy the other three axioms while failing the identity condition; natural-number addition fails the inverse condition; and a noncommutative group fails commutativity. Each example must be checked against all three retained axioms.