## Theorem                                        
                                                  
    Let G be a finite group of order 4.           
    Then either                                   
                                                  
        G ~= Z/4Z                                 
                                                  
    or                                            
                                                  
        G ~= Z/2Z x Z/2Z                          
                                                  
## Proof                                          
                                                  
The possibilities for the order of x in G with    
x != 1:                                           
                                                  
 1. If |x| = 4, then G = <x> such that x^4 = 1    
    => G = <x> ~= Z/4Z                            
    => G ~= Z/4Z                                  
                                                  
                                                  
 2. If |x| != 4, then |x| = 2 by Lagrange's       
    Theorem.  Since |G| = 4, there is a           
    y in G\{1,x} that has some order, |y|.        
                                                  
    Then if |y| = 4, we're done by (1). 
    Assume then that |y| != 4.  So then
    |y| = 2 by Lagrange.  We then have that
                                                  
        x, y in G                                 
        |x| = |y| = 2                             
        [G:<x>] = [G:<y>]                         
                = |G|/2                           
                = 2                               
                                                  
    Thus <x> and <y> are normal subgroups of      
    G since they have index 2.  Since <x>         
    intersect <y> = {1}, the map                  
                                                  
        p: <x> x <y> --> <x><y> < G               
                                                  
    is an isomorphism.                            
                                                  
    Since <x> ~= <y> ~= Z/2Z, we're done.         
                                                  
                                                  
                                      QED         
                                                  
                                                  
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Last Updated 2007-12-27                           
Ryan Timmons