PlanetMath: properties of group commutators and commutator subgroups

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properties of group commutators and commutator subgroups

(Theorem)

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

Let be a group.

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

Proof. Direct computation yields

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

$\qedsymbol$

Theorem 2 Let be subsets of , then .

Proof. By Theorem 1, the elements from

are products of commutators of the form

with $x\in X$ and $y\in Y$ . $\qedsymbol$

Theorem 3 (Hall-Witt identity) Let $x,y,z\in G$ , then

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

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

be the element obtained from

by the cyclic shift $S\colon x\mapsto y\mapsto z\mapsto x$ , and

be the element obtained from

. We have

$\displaystyle h_2^{-1}=(y^{-1}z^{-1}yx^{-1}y^{-1})^{-1}=yxy^{-1}zy$

which gives us

$\displaystyle y^{-1}[x,y^{-1},z]y=h_1h_2^{-1},$

and, by applying

twice

$\displaystyle z^{-1}[y,z^{-1},x]z$	$\displaystyle =h_2h_3^{-1},$
$\displaystyle x^{-1}[z,x^{-1},y]x$	$\displaystyle =h_3h_1^{-1}.$

In total, we have

$\displaystyle y^{-1}[x,y^{-1},z]yz^{-1}[y,z^{-1},x]zx^{-1}[z,x^{-1},y]x=h_1h_2^{-1}h_2h_3^{-1}h_3h_1^{-1}=1.$

$\qedsymbol$

Theorem 4 (Three subgroup lemma) Let be a normal subgroup of . Furthermore, let , and be subgroups of , such that and are contained in . Then is contained in as well.

Proof. The group

is generated by all elements of the form $[z,x^{-1},y]$ with $x\in X$ , $y\in Y$ and $z\in Z$ . Since

is normal, $y^{-1}[x,y^{-1},z]y$ and $x^{-1}[z,x^{-1},y]x$ are elements of

. The Hall-Witt identity then implies that $x^{-1}[z,x^{-1},y]x$ is an element of

as well. Again, since

is normal, $[z,x^{-1},y]\in N$ which concludes the proof. $\qedsymbol$

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 denotes $b^{-1} a b$

Proof. By expanding:

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

The other identities are proved similarly. $\qedsymbol$

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"properties of group commutators and commutator subgroups" is owned by GrafZahl. [ full author list (3) ]

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