identity element no relation commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, 
  
    
      
        G
        
          /
        
        N
      
    
    {\displaystyle G/N}
   is abelian if and only if 
  
    
      
        N
      
    
    {\displaystyle N}
   contains the commutator subgroup of 
  
    
      
        G
      
    
    {\displaystyle G}
  . So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.Commutators
For elements 
  
    
      
        g
      
    
    {\displaystyle g}
   and 
  
    
      
        h
      
    
    {\displaystyle h}
   of a group G, the commutator of 
  
    
      
        g
      
    
    {\displaystyle g}
   and 
  
    
      
        h
      
    
    {\displaystyle h}
   is 
  
    
      
        [
        g
        ,
        h
        ]
        =
        
          g
          
            −
            1
          
        
        
          h
          
            −
            1
          
        
        g
        h
      
    
    {\displaystyle [g,h]=g^{-1}h^{-1}gh}
  .  The commutator 
  
    
      
        [
        g
        ,
        h
        ]
      
    
    {\displaystyle [g,h]}
   is equal to the identity element e if and only if 
  
    
      
        g
        h
        =
        h
        g
      
    
    {\displaystyle gh=hg}
   , that is, if and only if 
  
    
      
        g
      
    
    {\displaystyle g}
   and 
  
    
      
        h
      
    
    {\displaystyle h}
   commute.  In general, 
  
    
      
        g
        h
        =
        h
        g
        [
        g
        ,
        h
        ]
      
    
    {\displaystyle gh=hg[g,h]}
  .
However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of the equation: 
  
    
      
        [
        g
        ,
        h
        ]
        =
        g
        h
        
          g
          
            −
            1
          
        
        
          h
          
            −
            1
          
        
      
    
    {\displaystyle [g,h]=ghg^{-1}h^{-1}}
   in which case  
  
    
      
        g
        h
        ≠
        h
        g
        [
        g
        ,
        h
        ]
      
    
    {\displaystyle gh\neq hg[g,h]}
   but instead 
  
    
      
        g
        h
        =
        [
        g
        ,
        h
        ]
        h
        g
      
    
    {\displaystyle gh=[g,h]hg}
  .
An element of G of the form 
  
    
      
        [
        g
        ,
        h
        ]
      
    
    {\displaystyle [g,h]}
   for some g and h is called a commutator.  The identity element e = [e,e] is always a commutator, and it is the only commutator if and only if G is abelian.
Here are some simple but useful commutator identities, true for any elements s, g, h of a group G:
no relation