= 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>.
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