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    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.

The eight motions I, R, R₁, R₂, H, V, D, D₁ can be represented as permutations of the numbers 1, 2, 3, and 4 where these numbers correspond to the four corners of the square as shown in Figure 1. Figure 3 shows their permutation representation. Thus G can also be represented as the set  

    G = [ (1), (1432), (13)(42), (1234), (14)(23), (12)(43), (42), (13) ]

of permutations of the vertices. In this case multiplication corresponds to the multiplication of permutations.

                                                Figure 3

Subgroups The subgroups of G are shown in Figure 4 along with their relationship to one another.

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Group G is not cyclic. It is generated by the two elements R and H. From the multiplication table it can be seen that

            R⁰ = I              R = R              R² = R₁           R³ = R₂

            H⁰ = I        H = H              HR = D₁         HR² = V         HR³ = D

Thus we see that the elements of G can be represented uniquely as HⁱR^j with i = 0,1 and

j = 0, 1, 2, 3 i.e.

            H⁰R⁰ = I         H⁰R¹ = R        H⁰R² = R₁      H⁰R³ = R₂

            H¹R⁰ = H        H¹R¹ = D₁      H¹R² = V        H¹R³ = D

Transforms. Figure 5 shows the transforms of each element a of G for each value of x. If we read across on the rows we see the conjugates of each element a. Thus the rows represent the different conjugate classes into which the group is partitioned. In some cases different rows give the same conjugate class. We can list the conjugate classes:

    class 1 = { I }

    class 2 = { R, R₂ }

    class 3 = { R₁ }

    class 4 = { H, V }

    class 5 = { D, D₁ }

These transforms correspond to automorphic mappings of the elements. Thus if we wish to know what subgroup a subgroup is mapped into by a particular automorphic mapping we can read it off from the table. For example, the subgroup [ I, H, V, R₁ ] is mapped into [ I, H, V, R₁ ] by the automorphic mapping T_R(a) : a x^-1ax i.e. it is mapped into itself.

References

   Birkhoff, Mac Lane. A Survey of Modern Algebra. Chap. VI