# Three subgroups lemma

(Redirected from Hall–Witt identity)

In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of the Hall–Witt identity.

## Notation

In that which follows, the following notation will be employed:

• If H and K are subgroups of a group G, the commutator of H and K[clarification needed] will be denoted by [H,K]; 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 CG(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, if ${\displaystyle N\triangleleft G}$, then 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, ${\displaystyle [z,x^{-1}]\subseteq {\mathbf {C}}_{G}(Y)}$ 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 ${\displaystyle [Z,X,Y]=1}$.