The Hall–Witt identity is the group-theoretic counterpart of the Jacobi identity in Lie algebras. It coordinates the three cyclic arrangements of a triple commutator and drives the three-subgroups lemma. [@isaacs2008]
Definition
One standard form is \([x,y^{-1},z]^y [y,z^{-1},x]^z [z,x^{-1},y]^x=1\). Opposite commutator conventions transform the entire formula.
Core result
A direct proof repeatedly expands with \([xy,z]=[x,z]^y[y,z]\) and \([x,yz]=[x,z][x,y]^z\) until all factors cancel. Isaacs, Sections 4A-4B gives the derivation used with the three-subgroups lemma.
Structural properties
- If two cyclic families of triple commutators lie in a normal subgroup \(N\), the third family does as well.
- Applying the identity to subgroups and taking normal closures yields the three-subgroups lemma.
- In a group of class at most \(2\), all triple commutators vanish and the identity becomes trivial.
- In the graded Lie ring associated with the lower central series, it induces the Jacobi identity.
- It is used to prove that distinct components commute and to control central commutator layers.
The identity is not merely a calculation formula; it is a structural cyclic symmetry connecting three directions of commutation.
Example and boundary
Knowledge network
The commutator core article fixes conventions, the three-subgroups lemma is the subgroup-level consequence, and the lower central series turns the identity into graded structure.
Proof and sources
The identity and expansion rules are in Sections 4A-4B. The long calculation is cited precisely rather than duplicated in this supporting entry. [@isaacs2008]
