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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 – 900 rotation (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

Tx(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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TR: 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

TR1: 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

TR2: 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

TH: 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

TV: V-1aV where a is any group member

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

V-1R1V = VR1V = HV = R1                                                            R1 → 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

TD: 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

TD1: 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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