Three subgroup lemma
This fact is related to: commutator calculus
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Contents
Statement
Two out of three formulation
Let be three subgroups of . Then any two of the three statements below implies the third:
Any one contained in normal closure of subgroup generated by other two
Let be three subgroups of . Then is contained in the normal closure of the subgroup generated by and . Equivalently, if is a normal subgroup containing both and , then contains .
Formulation where one is a group of automorphisms
Let be a group, be subgroups, and . Then, using the notation of commutator of element and automorphism, any two of the three statements below implies the third:
Further, is contained in the normal closure of the subgroup generated by and . Equivalently, if is a normal subgroup containing both and , then contains .
Proof
The three subgroup lemma follows from Witt's identity.
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Corollaries
If is a perfect group and is a subgroup of such that is trivial, then is trivial.
This result has an analogue in the theory of Lie algebras.
References
Textbook references
- Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, ^{More info}, Page 31, Theorem 2.1.2 (formal statement, with proof)