A group operation is the associative binary operation of a group. It has an identity element, and every element has an inverse element.
For a nonempty inclusion chain of group operations on subsets of a fixed set, the union is a group operation on the union of their carriers. Any finite list of carrier elements lies in one chain member, so its multiplication and associativity agree there. Nested identities coincide because the only idempotent of a group is its identity; inverses remain valid in larger members. Thus the union supplies the chain upper bound required by Zorn's lemma.
The group commutator measures the failure of two elements of a group to commute. Under the convention , it is the identity exactly when . The commutator subgroup is generated by these elements. The alternative convention generates the same commutator subgroup; individual formulas must specify their convention.
The Hall-Witt identity is a three-variable identity among iterated group commutators. Passing to the leading terms of a filtered group turns it into the Jacobi identity for the associated graded Lie algebra.
