The Dihedral Group of Order Eight

The Dihedral Group D4 of Order Eight

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

Last updates: 26 January 2011

The dihedral group D4 of order eight

The groups D4 is as in the following table.

Set Operation
D4= 1,x,x2,x3, y,xy,x2y,x3y xiyj xkyl= xi-kmod4 yj+lmod2

The complete multiplication tables for D4 is as follows.

Multiplication table
D4 1 x x2 x3 y xy x2y x3y
1 1 x x2 x3 y xy x2y x3y
x x x2 x3 1 xy x2y x3y y
x2 x2 x3 1 x x2y x3y y xy
x3 x3 1 x x2 x3y y xy x2y
y y x3y x2y xy 1 x3 x2 x
xy xy y x3y x2y x 1 x3 x2
x2y x2y xy y x3y x2 x 1 x3
x3y x3y x2y xy y x3 x2 x 1

Center Abelian Conjugacy classes Subgroups
ZD4= 1,x2 No 𝒞1 = 1 H0= D4
𝒞x2 = x2 H1= 1,x,x2,x3
𝒞y = y,x2y H2= 1,x2,y,x2y
𝒞xy = xy,x3y H3= 1,x2,xy,x3y
𝒞x = x,x3 H4= 1,x2
H5= 1,y
H6= 1,xy
H7= 1,x2y
H8= 1,x3y
H9= 1

Element g Order οg Centralizer Zg Conjugacy Class 𝒞g
1 1 D4 𝒞1
x 4 H1 𝒞x
x2 2 D4 𝒞x2
x3 4 H1 𝒞x
y 2 H2 𝒞y
xy 2 H3 𝒞xy
x2y 2 H2 𝒞y
x3y 2 H3 𝒞xy

Generators Relations
D4 x,y x4=y2=1
yx=x-1y

Subgroups Hi Structure Index Normal Quotient group
H0= D4 H0=D4 D4:D4 = 1 Yes D4 /H0 1
H1= 1,x,x2,x3 H1 C4 D4:H1 = 2 Yes D4 /H1 C2
H2= 1,x2,y,x2y H2 C2×C2 D4:H2 =2 Yes D4 /H2 C2
H3= 1,x2,xy,x3y H3 C2×C2 D4:H3 =2 Yes D4 /H3 C2
H4= 1,x2 H4C2 D4:H4 =4 No
H5= 1,y H5C2 D4:H5 =4 No
H6= 1,xy H6C2 D4:H6 =4 No
H7= 1,x2y H7C2 D4:H7 =4 No
H8= 1,x3y H8C2 D4:H8 =4 No
H9= 1 H9=1 D4:1 =1 Yes D4 /H9 D4

Orders Inclusions 8 4 2 1 D4 x= 1,x,x2,x3 1,x2,y,x2y 1,x2,xy,x3y x2= 1,x2 y= 1,y x2y= 1,x2y xy= 1,xy x3y= 1,x3y 1

Subgroups Hi Left Cosets Right Cosets
H0=D4 D4=xD4 =x3D4=yD4 D4=D4x= D4x2= D4x3 =D4y
=xyD4 =x2yD4 =x3yD4 = D4xy= D4x2y= D4x3y
H1 = 1,x,x2,x3 H1 =xH1=x2H1 =x3H1 H1=H1x= H1x2=H1x3
= 1,x,x2,x3 = 1,x,x2,x3
yH1=xyH1 =x2yH1= x3yH1 H1y=H1xy= H1x2y= H1x3y
= y,xy,x2y,x3y = y,xy,x2y,x3y
H2= 1,x2,y,x2y H2=x2H2 =yH2=x2yH2 H2=H2x2 =H2y=H2 x2y
= 1,x2,y,x2y = 1,x2,y,x2y
xH2=x3H2= xyH2= x3yH2 H2x=H2x3= H2xy=H2x3 y
= x,x3,xy,x3y = x,x3,xy,x3y
H3= 1,x2,xy,x3y H3=x2H3 =xyH3=x3y H3 H3=H3x2 =H3xy=H3 x3y
= 1,x2,xy,x3y = 1,x2,xy,x3y
xH3=x3H3 =yH3=x2y H3 H3x=H3x3= H3y=H3x2y
= x,x3,y,x2y = x,x3,y,x2y
H4= 1,x2 H4=x2H4 = 1,x2 H4=H4x2 = 1,x2
xH4=x3H4 = x,x3 H4x=H4x3 = x,x3
yH4=x2yH4 = y,x2y H4y=H4x2y = y,x2y
xyH4= x3yH4= xy,x3y H4xy= H4x3y= xy,x3y
H5= 1,y H5=yH5 = 1,y H5=H5y = 1,y
xH5= xyH5= x,xy H5x= H5x3y= x,x3y
x2H5= x2yH5= y,x2y H5x2= H5x2y= x2,x2y
x3H5= x3yH5= x3,x3y H5x3= H5xy= x3,xy
H6= 1,xy H6=xyH6= 1,xy H6=H6xy= 1,xy
xH6=x2yH6 = x,x2y H6x=H6x2y = x,y
x2H6 =x3yH6 = x2,x3y H6x2 =H6x3y = x2,x3y
x3H6 =yH6 = x3,y H6x3 =H6x2y = x3,x2y
H7= 1,x2y H7= x2yH7= x,x2y H7= H7x2y = 1,x2y
xH7= x3yH7= x,x3y H7x= H7xy= x,xy
x2H7= yH7= x2,y H7x2= H7y= x2,y
x3H7= xyH7= x3,xy H7x3= H7x3y= x3,x3y
H8= 1,x3y H8=x3yH8 = 1,x3y H8=H8x3y= 1,x3y
xH8=yH8= x,y H8x=H8x2y= x2y,y
x2H8= xyH8= x2,xy H8x2= H8xy= x2,xy
x3H8= x2yH8 = x3,x2y H8x3=H8y= x3,y
H9= 1 H9= 1 , xH9= x , H9= 1 , H9x= x ,
x2H9= x2 , xH9= x , H9x2= x2 , H9x3= x3 ,
yH9= y , xyH9= xy , H9y= y , H9xy= xy ,
x2yH9= x2y , x3yH9= x3y , H9x2y= x2y , H9x3y= x3y ,

Subgroups Hi Normalizer NHi Centralizer ZHi
H0=D4 H0=D4 ZD4= H4= x2
H1= x D4 H1= x
H2= x2,y D4 H2= x2,y
H3= x2,xy D4 H3= x2,xy
H4= x2 D4 D4
H5= y H2= x2,y H2= x2,y
H6= xy H3= x2,xy H3= x2,xy
H7= x2y H2= x2,y H2= x2,y
H8= x3y H3= x2,xy H3= x2,xy
H9= 1 D4 D4

Homomorphism Kernel Image
ϕ0: D4 1 s1 1 s2 1 kerϕ0= D4 imϕ0=1
ϕ1: D4 μ2 x 1 y -1 kerϕ1= H1 imϕ1 =μ2
ϕ2: D4 μ2 x -1 y 1 kerϕ2= 1,x2,y,x2y =H2 imϕ2 =μ2
ϕ3: D4 μ2 x -1 y -1 kerϕ3= 1,x2,xy,x3y =H2 imϕ3 =μ2
ϕ4: D4 μ2×μ2 x -1 1 y 1 -1 kerϕ4= 1,x2 =H4 imϕ4 =μ2×μ2
ϕ9: D4 D4 x x y y kerϕ3= 1 =H9 imϕ9 =D4

References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

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

page history