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 [H, K], 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] = [[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 [math]\displaystyle{ x^{y} }[/math].
• If H is a subgroup of a group G, then the centralizer of H in G will be denoted by C_G(H).

Statement

Let X, Y and Z be subgroups of a group G, and assume

[math]\displaystyle{ [X,Y,Z]=1 }[/math] and [math]\displaystyle{ [Y,Z,X]=1. }[/math]

Then [math]\displaystyle{ [Z,X,Y]=1 }[/math].

More generally, for a normal subgroup [math]\displaystyle{ N }[/math] of [math]\displaystyle{ G }[/math], if [math]\displaystyle{ [X,Y,Z]\subseteq N }[/math] and [math]\displaystyle{ [Y,Z,X]\subseteq N }[/math], then [math]\displaystyle{ [Z,X,Y]\subseteq N }[/math].

Proof and the Hall–Witt identity

Hall–Witt identity

If [math]\displaystyle{ x,y,z\in G }[/math], then

[math]\displaystyle{ [x, y^{-1}, z]^y\cdot[y, z^{-1}, x]^z\cdot[z, x^{-1}, y]^x = 1. }[/math]

Proof of the three subgroups lemma

Let [math]\displaystyle{ x\in X }[/math], [math]\displaystyle{ y\in Y }[/math], and [math]\displaystyle{ z\in Z }[/math]. Then [math]\displaystyle{ [x,y^{-1},z]=1=[y,z^{-1},x] }[/math], and by the Hall–Witt identity above, it follows that [math]\displaystyle{ [z,x^{-1},y]^{x}=1 }[/math] and so [math]\displaystyle{ [z,x^{-1},y]=1 }[/math]. Therefore, [math]\displaystyle{ [z,x^{-1}]\in \mathbf{C}_G(Y) }[/math] for all [math]\displaystyle{ z\in Z }[/math] and [math]\displaystyle{ x\in X }[/math]. Since these elements generate [math]\displaystyle{ [Z,X] }[/math], we conclude that [math]\displaystyle{ [Z,X]\subseteq \mathbf{C}_G(Y) }[/math] and hence [math]\displaystyle{ [Z,X,Y]=1 }[/math].

See also

• Commutator
• Lower central series
• Grün's lemma
• Jacobi identity

References

• I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.