Dihedral Groups

Dihedral Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 6 December 2010

Definition. The Dihedral group, Dn is the set Dn= 1, x, x2, , xn-1, y, xy, x2y, , xn-1y with the operation given by xi yj xk yl = x i+kmodn y j+lmod2 .

HW: Show that the order of the dihedral group Dn is 2n.

The orders of the elements in the dihedral group Dn are ο1=1, οxk= gcdk,n, οxky=2, 0<kn-1.

Conjugacy classes, normal subgroups, and the center

  1. The conjugacy classes of the dihedral group D2 are the sets 𝒞1= 1 , 𝒞x= x , 𝒞y= y , 𝒞xy= xy .
  2. If n is even and n2, then the conjugacy classes of the dihedral group Dn are the sets 𝒞1= 1 , 𝒞xn/2 = xn/2 , 𝒞xk= xk,x-k , 0<k<n/2, 𝒞y= y, x2y, x4y, , xn-2y , 𝒞xy= xy, x3y, x5, , xn-1y .
  3. If n is even and n2, then the conjugacy classes of the dihedral group Dn are the sets 𝒞1= 1 , 𝒞xk= xk,x-k , 0<k<n/2, 𝒞y= y, xy, x2y, x3y, , xn-1y .

Let a,b, denote the subgroup generated by the elements a,b,.

  1. The normal subgroups of the dihedral group D2 are the subgroups x, y, xy.
  2. If n is even and n2 then the normal subgroups of the dihedral group Dn are the subgroups xk, 0kn-1, x2,y, x2,xy.
  3. If n is odd, then the normal subgroups of the dihedral group Dn are the subgroups xk, 1kn-1.

  1. The center of the dihedral group D2 is the subgroup ZD2=D2.
  2. If n is even and n2,then the center of the dihedral group Dn is the subgroup ZDn= 1,xn/2 .
  3. If n is odd, then the center of the dihedral group Dn is the subgroup ZDn= 1.

The action of Dn on an n-gon

Let F be an n-gon with vertices v0, v1, , vn-1 numbered counterclockwise around F. Then there is an action of the group Dn on the n-gon F such that

  1. x acts by rotating the n-gon by an angle of 2π/n;
  2. y acts by reflecting about the line which contains the vertex v0 and the center of F.

v0 v5 v4 v3 v2 v1 y x

Generators and relations

  1. The dihedral group Dn= 1, x, x2, , xn-1, y, yx, yx2, , yxn-1 is generated by the elements x and y.
  2. The elements x and y in Dn satisfy the relations xn=1, y2=1, yx=x-1y.

The dihedral group Dn has a presentation by generators and relations by Dn= x,y xn=1, y2=1, yx=x-1y.

References

[Bou] N. Bourbaki, Groupes et Alg├Ębres de Lie, Masson, Paris, 1990.

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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