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[parent] 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 $ G$ 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 $ 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$. $ \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 $ 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
$\displaystyle h_2^{-1}=(y^{-1}z^{-1}yx^{-1}y^{-1})^{-1}=yxy^{-1}zy$    

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

and, by applying $ S$ 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 $ 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. $ \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 $ a^b$ 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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See Also: normal subgroup

Also defines:  Hall-Witt identity, three subgroup lemma

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Cross-references: implies, generated by, contained, normal subgroup, subgroup, cyclic, factor, calculate, identity, commutators, products, subsets, group, commutator subgroups, properties
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This is version 8 of properties of group commutators and commutator subgroups, born on 2005-09-21, modified 2007-12-15.
Object id is 7381, canonical name is PropertiesOfGroupCommutatorsAndCommutatorSubgroups.
Accessed 3367 times total.

Classification:
AMS MSC20F12 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Commutator calculus)

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