Computation of the conjugate table for The Dihedral Group of the Square

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Computation of the conjugate table for the Dihedral Group of the Square

Consider a cardboard square as shown in Figure 1. There are eight motions of this square which, when performed one after the other, form a group called the Dihedral Group of the Square. They are:

I – 0⁰ rotation (clockwise, about center O, in plane of cardboard)

R – 90⁰rotation (clockwise, about center O, in plane of cardboard)

R₁ – 180⁰ rotation (clockwise, about center O, in plane of cardboard)

R₂ – 270⁰ rotation (clockwise, about center O, in plane of cardboard)

H – reflection about horizontal axis AB (180⁰ flip through space)

V – reflection about vertical axis EF (180⁰ flip through space)

D – reflection about diagonal 1-O-3 (180⁰ flip through space)

D₁ – reflection about diagonal 2-O-4 (180⁰ flip through space)

The Dihedral Group of the Square then is given by G = [ I, R, R₁, R₂, H, V, D, D₁ ]. Multiplication in G consists of performing two of these motions in succession. Thus the product HR corresponds to first performing operation H, then operation R. A multiplication table for G is shown in Figure 2. Entries in the table contain the product XY where X corresponds to the row and Y corresponds to the column. Thus in the table HR = D₁.

Utilizing the multiplication table of Table 2, we now compute the transforms

T_x(a) : a → x ^-1ax

of each member a ∈ G for all elements x ∈ G to construct the table of transforms shown in Figure 5. The table of transforms shown in Fig. 5 displays the various conjugate classes. Each line of the table represents a set of elements that are conjugate to each other i.e. a particular conjugate class.

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T_R: R^-1xR where x is any group member

R^-1RR = R (R^-1 = R₂) R → R

R^-1R₁R = R₂R₁R = RR = R₁ R₁ →R₁

R^-1R₂R = R₂R₂R = R₁R = R₂ R₂ →R₂

R^-1HR = R₂HR = D₁R = V H → V

R^-1VR = R₂VR = DR = H V → H

R^-1DR = R₂DR = HR = D₁ D → D₁

R^-1D₁R = R₂D₁R = VR = D D₁ → D

T_R1: R₁^-1aR₁ where a is any group member

R₁^-1RR₁ = R (R₁^-1 = R₁) R → R

R₁^-1R₁R₁ = R₁ R₁ →R₁

R₁^-1R₂R₁ = R₁R₂R₁ = RR₁ = R₂ R₂ →R₂

R₁^-1HR₁ = R₁HR₁ = VR₁ = H H → H

R₁^-1VR₁ = R₁VR₁ = HR₁ = V V → V

R₁^-1DR₁ = R₁DR₁ = VD₁R₁ = D D → D

R₁^-1D₁R₁ = R₁D₁R₁ = DR₁ = D₁ D₁ → D₁

T_R2: R₂^-1aR₂ where a is any group member

R₂^-1RR₂ = RRR₂ = R₁R₂ = R (R₂^-1 = R) R → R

R₂^-1R₁R₂ = RR₁R₂ = R₂R₂ = R₁ R₁ →R₁

R₂^-1R₂R₂ = R₂ R₂ →R₂

R₂^-1HR₂ = RHR₂ = DR₂ = V H → V

R₂^-1VR₂ = RVR₂ = D₁R₂ = H V → H

R₂^-1DR₂ = RDR₂ = VR₂ = D₁ D → D₁

R₂^-1D₁R₂ = RD₁R₂ = HR₂ = D D₁ → D

T_H: H^-1aH where a is any group member

H^-1RH = HRH = D₁H = R₂ (H^-1 = H) R → R₂

H^-1R₁H = HR₁H = VH = R₁ R₁ →R₁

H^-1R₂H = HR₂H = DH = R R₂ → R

H^-1HH = H H → H

H^-1VH = HVH = R₁H = V V → V

H^-1DH = HDH = R₂H = D₁ D → D₁

H^-1D₁H = HD₁H = RH = D D₁ → D

T_V: V^-1aV where a is any group member

V^-1RV = VRV = DV = R₂ (V^-1 = V) R → R₂

V^-1R₁V = VR₁V = HV = R₁R₁→ R₁

V^-1 R₂V = VR₂V = D₁V = R R₂ → R

V^-1HV = VHV = R₁V = H H → H

V^-1VV = V V → V

V^-1DV = VDV = RV = D₁ D → D₁

V^-1D₁V = VD₁V = R₂V = D D₁ → D

T_D: D^-1aD where a is any group member

D^-1RD = DRD = HD = R₂ (D^-1 = D) R → R₂

D^-1R₁D = DR₁D = D₁D = R₁ R₁ → R₁

D^-1R₂D = DR₂D = VD = R R₂→ R

D^-1HD = DHD = RD = V H → V

D^-1VD = DVD = R₂D = H V → H

D^-1DD = D D → D

D^-1D₁D = DD₁D = R₁D = D₁ D₁ →D₁

T_D1: D₁^-1aD₁ where a is any group member

D₁^-1RD₁ = D₁RD₁ = VD₁ = R₂ (D₁^-1 = D₁) R → R₂

D₁^-1R₁D₁ = D₁R₁D₁ = DD₁ = R₁ R₁ → R₁

D₁^-1R₂D₁ = D₁R₂D₁ = HD₁ = R R₂→ R

D₁^-1HD₁ = D₁HD₁ = R₂D₁ = V H → V

D₁^-1VD₁ = D₁VD₁ = RD₁ = H V → H

D₁^-1DD₁ = D₁DD₁ = R₁D₁ = D D → D

D₁^-1D₁D₁ == D₁ D₁ →D₁

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