Three subgroups lemma
In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.
Notation
In what follows, the following notation will be employed:- If H and K are subgroups of a group G, the commutator of H and K, denoted by, is defined as the subgroup of G generated by commutators between elements in the two subgroups. If L is a third subgroup, the convention that = H,''K],L''] will be followed.
- If x and y are elements of a group G, the conjugate of x by y will be denoted by.
- If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG.
Statement
Let X, Y and Z be subgroups of a group G, and assumeThen.
More generally, for a normal subgroup of, if and, then.
Proof and the Hall–Witt identity
Hall-Witt identityIf, then
Proof of the three subgroups lemma
Let,, and. Then, and by the Hall-Witt identity above, it follows that and so. Therefore, for all and. Since these elements generate, we conclude that and hence.