Binary cyclic group

<h2 id="presentation">Presentation</h2>
<p>The <i>binary cyclic group</i> can be defined as the set of 
  
    
      
        2
        n
      
    
    {\displaystyle 2n}
  
th roots of unity—that is, the set 
  
    
      
        
          {
          
            
              ω
              
                n
              
              
                k
              
            
            
            
              |
            
            
            k
            ∈
            {
            0
            ,
            1
            ,
            2
            ,
            .
            .
            .
            ,
            2
            n
            −
            1
            }
          
          }
        
      
    
    {\displaystyle \left\{\omega _{n}^{k}\;|\;k\in \{0,1,2,...,2n-1\}\right\}}
  
, where
</p>

ω
          
            n
          
        
        =
        
          e
          
            i
            π
            
              /
            
            n
          
        
        =
        cos
        ⁡
        
          
            π
            n
          
        
        +
        i
        sin
        ⁡
        
          
            π
            n
          
        
        ,
      
    
    {\displaystyle \omega _{n}=e^{i\pi /n}=\cos {\frac {\pi }{n}}+i\sin {\frac {\pi }{n}},}

<p>using multiplication as the group operation.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Binary_dihedral_group/nLIvYLD6">binary dihedral group</a>, ⟨2,2,<i>n</i>⟩, order 4<i>n</i></li>
<li><a href="/facts/Binary_tetrahedral_group/pBBUSr2g">binary tetrahedral group</a>, ⟨2,3,3⟩, order 24</li>
<li><a href="/facts/Binary_octahedral_group/0MaAuuWa">binary octahedral group</a>, ⟨2,3,4⟩, order 48</li>
<li><a href="/facts/Binary_icosahedral_group/9JmfJ4Rc">binary icosahedral group</a>, ⟨2,3,5⟩, order 120</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Coxeter, H. S. M. (1959), "Symmetrical definitions for the binary polyhedral groups", Proc. Sympos. Pure Math., Vol. 1, Providence, R.I.: American Mathematical Society, pp. 64–87, MR 0116055. <a href="/wiki/Harold_Scott_MacDonald_Coxeter" target="_blank">/wiki/Harold_Scott_MacDonald_Coxeter</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Binary cyclic group open-in-new

Binary cyclic group