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 ${\displaystyle x^{y}}$.
• 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

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

Then ${\displaystyle [Z,X,Y]=1}$.^[1]

More generally, for a normal subgroup ${\displaystyle N}$ of ${\displaystyle G}$, if ${\displaystyle [X,Y,Z]\subseteq N}$ and ${\displaystyle [Y,Z,X]\subseteq N}$, then ${\displaystyle [Z,X,Y]\subseteq N}$.^[2]

Proof and the Hall–Witt identity

Hall–Witt identity

If ${\displaystyle x,y,z\in G}$, then

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

Proof of the three subgroups lemma

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

See also

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