# Three subgroups lemma

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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

HallWitt 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 HallWitt identity above, it follows that $[z,x^{-1},y]^{x}=1$ and so $[z,x^{-1},y]=1$ . Therefore, $[z,x^{-1}]\subseteq \mathbf {C} _{G}(Y)$ 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 $[Z,X,Y]=1$ .