The three-subgroups lemma promotes the elementwise Hall–Witt identity to a subgroup-level inference rule. It allows two of three cyclic commutator conditions to determine the third. [@isaacs2008]
Definition
A common use is first to quotient by \(N\), reduce all inclusions to equality with \(1\), and apply the lemma there.
Core result
After quotienting by \(N\), assume two triple commutator subgroups are trivial. Apply Hall–Witt to individual elements, then use normal closure and generation to obtain the subgroup conclusion.
Structural properties
- Taking \(N=1\) gives the usual centralization form.
- If \([X,Y]\) centralizes \(Z\) and \([Y,Z]\) centralizes \(X\), then \([Z,X]\) centralizes \(Y\).
- The lemma eliminates residual central commutators when proving that quasisimple components commute.
- It is also used in comparing central series and proving commutator-weight inclusions.
- Cyclic permutation of the three variables is built in; arbitrary swaps require the relevant inverse identities.
This is a local 'two faces determine the third' principle that often collapses lengthy commutator calculations.
Example and boundary
Knowledge network
Hall–Witt supplies the element identity, the commutator core article fixes notation, and generalized Fitting theory exhibits the component-commuting application.
Proof and sources
The subgroup lemma and its Hall–Witt derivation are in Section 4B, with applications throughout Chapters 4 and 9. [@isaacs2008]
