Modular group

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the modular group  is the <a href="/facts/Projective_linear_group/y7P9rMyR">projective special linear group</a> 
  
    
      
        PSL
        ⁡
        (
        2
        ,
        
          Z
        
        )
      
    
    {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}
  
 of 
  
    
      
        2
        ×
        2
      
    
    {\displaystyle 2\times 2}
  
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> with <a href="/facts/Integer/PUwwrawx">integer</a> coefficients and <a href="/facts/Determinant/eGYqJ2TF">determinant</a> 
  
    
      
        1
      
    
    {\displaystyle 1}
  
, such that the matrices 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        −
        A
      
    
    {\displaystyle -A}
  
 are identified. The modular group acts on the upper-half of the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> by <a href="/facts/Linear_fractional_transformation/nykA54lI">linear fractional transformations</a>. The name "modular group" comes from the relation to <a href="/facts/Moduli_space/9sqrdpwR">moduli spaces</a>, and not from <a href="/facts/Modular_arithmetic/0wtRa5dU">modular arithmetic</a>.
</p>

Modular group open-in-new

Modular group