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 $$x^{y}$$.
 * If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG(H).

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


 * $$[X,Y,Z]=1$$ and $$[Y,Z,X]=1.$$

Then $$[Z,X,Y]=1$$.

More generally, for a normal subgroup $$N$$ of $$G$$, if $$[X,Y,Z]\subseteq N$$ and $$[Y,Z,X]\subseteq N$$, then $$[Z,X,Y]\subseteq N$$.

Proof and the Hall–Witt identity
Hall–Witt identity

If $$x,y,z\in G$$, then


 * $$[x, y^{-1}, z]^y\cdot[y, z^{-1}, x]^z\cdot[z, x^{-1}, y]^x = 1.$$

Proof of the three subgroups lemma

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