Three subgroup lemma

This fact is related to: commutator calculus
View other facts related to commutator calculus | View terms related to commutator calculus

This article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic
View other semi-basic facts in group theory
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
VIEW: Survey articles about this

Statement

Symbolic statement

Let $A,B,C$ be three subgroups of $G$. Then any two of the three statements below implies the third:

• $[[A,B],C]$ is trivial
• $[[B,C],A]$ is trivial
• $[[C,A],B]$ is trivial

Proof

The three subgroup lemma follows from Witt's identity.

Corollaries

If $G$ is a perfect group and $N$ is a subgroup of $G$ such that $[[G,N],N]$ is trivial, then $[G,N]$ is trivial.

This result has an analogue in the theory of Lie algebras.