R
        
      
    
    {\displaystyle \mathbb {R} }
   has a vector space of dimension 1, while 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  , 
  
    
      
        
          H
        
      
    
    {\displaystyle \mathbb {H} }
  , 
  
    
      
        
          O
        
      
    
    {\displaystyle \mathbb {O} }
   and 
  
    
      
        
          S
        
      
    
    {\displaystyle \mathbb {S} }
   work in algebraic number fields of dimensions 2, 4, 8, and 16, respectively.
An alternating groups are not simple for values 
  
    
      
        n
      
    
    {\displaystyle n}
   ≤ 
  
    
      
        4
      
    
    {\displaystyle 4}
  .