View source for Three subgroup lemma

{{factrelatedto|commutator calculus}}

{{semibasic fact}}

==Statement==

===Two out of three formulation===

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

* <math>[[A,B],C]</math> is [[trivial group|trivial]]
* <math>[[B,C],A]</math> is [[trivial group|trivial]]
* <math>[[C,A],B]</math> is [[trivial group|trivial]]

===Any one contained in normal closure of subgroup generated by other two===

Let <math>A, B, C</math> be three subgroups of <math>G</math>. Then <math>[[A,B],C]</math> is contained in the [[normal closure]] of the subgroup generated by <math>[[B,C],A]</math> and <math>[[C,A],B]</math>. Equivalently, if <math>N</math> is a normal subgroup containing both <math>[[B,C],A]</math> and <math>[[C,A],B]</math>, then <math>N</math> contains <math>[[A,B],C]</math>.

===Formulation where one is a group of automorphisms===

Let <math>G</math> be a group, <math>A,B</math> be subgroups, and <math>C \le \operatorname{Aut}(G)</math>. Then, using the notation of [[commutator of element and automorphism]], any two of the three statements below implies the third:

* <math>[[A,B],C]</math> is trivial
* <math>[[B,C],A]</math> is [[trivial group|trivial]]
* <math>[[C,A],B]</math> is [[trivial group|trivial]]

Further, <math>[[A,B],C]</math> is contained in the [[normal closure]] of the subgroup generated by <math>[[B,C],A]</math> and <math>[[C,A],B]</math>. Equivalently, if <math>N</math> is a normal subgroup containing both <math>[[B,C],A]</math> and <math>[[C,A],B]</math>, then <math>N</math> contains <math>[[A,B],C]</math>.
==Proof==

The three subgroup lemma follows from [[Witt's identity]].

{{fillin}}

==Corollaries==

If <math>G</math> is a [[perfect group]] and <math>N</math> is a subgroup of <math>G</math> such that <math>[[G,N],N]</math> is trivial, then <math>[G,N]</math> is trivial.

This result has an analogue in the theory of [[Lie algebra]]s.

==References==
===Textbook references===
* {{booklink-proved|KhukhroNGA}}, Page 31, Theorem 2.1.2 (formal statement, with proof)