Three subgroups lemma explained

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 C_G(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

.^[1]

, if

[X,Y,Z]\subseteqN

and

[Y,Z,X]\subseteqN

, then

[Z,X,Y]\subseteqN

.^[2]

Proof and the Hall - Witt identity

Hall - Witt identity

x,y,z\inG

, then

[x,y^-1,z]^{y ⋅ [y,}z^-1,x]^{z ⋅ [z,}x^-1,y]^x=1.

Proof of the three subgroups lemma

Let

x\inX

y\inY

, and

z\inZ

. 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]\inC_G(Y)

for all

z\inZ

and

x\inX

. Since these elements generate

[Z,X]

, we conclude that

[Z,X]\subseteqC_G(Y)

and hence