properties of group commutators and commutator subgroups

The purpose of this entry is to collect properties of commutators and commutator subgroupsMathworldPlanetmath. Feel free to add more theorems!

Let G be a group.

Theorem 1.

Let x,yG, then [x,y]-1=[y,x].


Direct computation yields


Theorem 2.

Let X,Y be subsets of G, then [X,Y]=[Y,X].


By Theorem 1, the elements from [X,Y] or [Y,X] are products of commutators of the form [x,y] or [y,x] with xX and yY. ∎

Theorem 3 (Hall–Witt identity).

Let x,y,zG, then


This is mainly a brute-force calculation. We can easily calculate the first factor y-1[x,y-1,z]y explicitly using theorem 1:

= y-1[y-1,x]z-1[x,y-1]zy
= y-1yx-1y-1xz-1x-1yxy-1zy
= x-1y-1xz-1x-1yxy-1zy.

Let h1:=x-1y-1xz-1x-1, the “first half” of y-1[x,y-1,z]y. Let h2 be the element obtained from h1 by the cyclic shift S:xyzx, and h3 be the element obtained from h2 by S. We have


which gives us


and, by applying S twice

z-1[y,z-1,x]z =h2h3-1,
x-1[z,x-1,y]x =h3h1-1.

In total, we have


Theorem 4 (Three subgroup lemma).

Let N be a normal subgroupMathworldPlanetmath of G. Furthermore, let X, Y and Z be subgroupsMathworldPlanetmathPlanetmath of G, such that [X,Y,Z] and [Y,Z,X] are contained in N. Then [Z,X,Y] is contained in N as well.


The group [Z,X,Y] is generated by all elements of the form [z,x-1,y] with xX, yY and zZ. Since N is normal, y-1[x,y-1,z]y and x-1[z,x-1,y]x are elements of N. The Hall–Witt identityPlanetmathPlanetmath then implies that x-1[z,x-1,y]x is an element of N as well. Again, since N is normal, [z,x-1,y]N which concludes the proof. ∎

Theorem 5.

For any x,y,zG we have

[xy,z] = [x,z]y[y,z]
[x,yz] = [x,z][x,y]z
[x,y]z = [xz,yz]
[xz,y] = [x,yz-1]

where ab denotes b-1ab


By expanding:

[xy,z] = y-1x-1z-1xyz
= y-1x-1z-1xzz-1x-1xyz
= y-1[x,z]yy-1z-1x-1xyz
= [x,z]yy-1z-1yz
= [x,z]y[y,z]

The other identities are proved similarly. ∎

Title properties of group commutators and commutator subgroups
Canonical name PropertiesOfGroupCommutatorsAndCommutatorSubgroups
Date of creation 2013-03-22 15:30:50
Last modified on 2013-03-22 15:30:50
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 11
Author GrafZahl (9234)
Entry type Theorem
Classification msc 20F12
Related topic NormalSubgroup
Defines Hall-Witt identity
Defines three subgroup lemma