properties of group commutators and commutator subgroups

The purpose of this entry is to collect properties of http://planetmath.org/node/2812group commutators and commutator subgroups. Feel free to add more theorems!

Let $G$ be a group.

Theorem 1.

Let $x,y\in G$, then $[x,y]^{-1}=[y,x]$.

Proof.

Direct computation yields

 $[x,y]^{-1}=(x^{-1}y^{-1}xy)^{-1}=y^{-1}x^{-1}yx=[y,x].$

Theorem 2.

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

Proof.

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

Theorem 3 (Hall–Witt identity).

Let $x,y,z\in G$, then

 $y^{-1}[x,y^{-1},z]yz^{-1}[y,z^{-1},x]zx^{-1}[z,x^{-1},y]x=1.$
Proof.

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:

 $\displaystyle y^{-1}[x,y^{-1},z]y$ $\displaystyle=$ $\displaystyle y^{-1}[y^{-1},x]z^{-1}[x,y^{-1}]zy$ $\displaystyle=$ $\displaystyle y^{-1}yx^{-1}y^{-1}xz^{-1}x^{-1}yxy^{-1}zy$ $\displaystyle=$ $\displaystyle x^{-1}y^{-1}xz^{-1}x^{-1}yxy^{-1}zy.$

Let $h_{1}:=x^{-1}y^{-1}xz^{-1}x^{-1}$, the “first half” of $y^{-1}[x,y^{-1},z]y$. Let $h_{2}$ be the element obtained from $h_{1}$ by the cyclic shift $S\colon x\mapsto y\mapsto z\mapsto x$, and $h_{3}$ be the element obtained from $h_{2}$ by $S$. We have

 $h_{2}^{-1}=(y^{-1}z^{-1}yx^{-1}y^{-1})^{-1}=yxy^{-1}zy$

which gives us

 $y^{-1}[x,y^{-1},z]y=h_{1}h_{2}^{-1},$

and, by applying $S$ twice

 $\displaystyle z^{-1}[y,z^{-1},x]z$ $\displaystyle=h_{2}h_{3}^{-1},$ $\displaystyle x^{-1}[z,x^{-1},y]x$ $\displaystyle=h_{3}h_{1}^{-1}.$

In total, we have

 $y^{-1}[x,y^{-1},z]yz^{-1}[y,z^{-1},x]zx^{-1}[z,x^{-1},y]x=h_{1}h_{2}^{-1}h_{2}% h_{3}^{-1}h_{3}h_{1}^{-1}=1.$

Theorem 4 (Three subgroup lemma).

Let $N$ be a normal subgroup of $G$. Furthermore, let $X$, $Y$ and $Z$ be subgroups 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.

Proof.

The group $[Z,X,Y]$ is generated by all elements of the form $[z,x^{-1},y]$ with $x\in X$, $y\in Y$ and $z\in Z$. 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 identity then implies that $x^{-1}[z,x^{-1},y]x$ is an element of $N$ as well. Again, since $N$ is normal, $[z,x^{-1},y]\in N$ which concludes the proof. ∎

Theorem 5.

For any $x,y,z\in G$ we have

 $\displaystyle[xy,z]$ $\displaystyle=$ $\displaystyle[x,z]^{y}[y,z]$ $\displaystyle{[}x,yz]$ $\displaystyle=$ $\displaystyle[x,z][x,y]^{z}$ $\displaystyle{[}x,y]^{z}$ $\displaystyle=$ $\displaystyle[x^{z},y^{z}]$ $\displaystyle{[}x^{z},y]$ $\displaystyle=$ $\displaystyle[x,y^{z^{-1}}]$

where $a^{b}$ denotes $b^{-1}ab$

Proof.

By expanding:

 $\displaystyle[xy,z]$ $\displaystyle=$ $\displaystyle y^{-1}x^{-1}z^{-1}xyz$ $\displaystyle=$ $\displaystyle y^{-1}x^{-1}z^{-1}\cdot xz\cdot z^{-1}x^{-1}\cdot xyz$ $\displaystyle=$ $\displaystyle y^{-1}[x,z]\cdot y\cdot y^{-1}\cdot z^{-1}x^{-1}\cdot xyz$ $\displaystyle=$ $\displaystyle[x,z]^{y}\cdot y^{-1}z^{-1}yz$ $\displaystyle=$ $\displaystyle[x,z]^{y}[y,z]$

The other identities are proved similarly. ∎

Title properties of group commutators and commutator subgroups PropertiesOfGroupCommutatorsAndCommutatorSubgroups 2013-03-22 15:30:50 2013-03-22 15:30:50 GrafZahl (9234) GrafZahl (9234) 11 GrafZahl (9234) Theorem msc 20F12 NormalSubgroup Hall-Witt identity three subgroup lemma