D4---Group, Subgroups and Cosets
Materials and Methods: Cross-stitch
Designer & Artist: Mary Shepherd
The group D4 can be thought of as the symmetry transformations of a square. The diagram below encodes the word description that follows. The full group is represented by the full flower motifs in the upper left and lower right corners. Each of the eight elements of D4, represented by one of the flowers with a single filled petal, appears along the left or right edge of the table; the identity element is the first single petal flower in the second row. Each single petal flower represents the symmetry transformation of D4 that takes the petal representing the identity element to that specific petal. Thus, the single petal flower at the left end of the third row represents the symmetry transformation reflect over the vertical midline, and the single petal flower below that represents the symmetry transformation rotate 90 degrees clockwise. Across the top and bottom rows, excluding the left and right ends, appear the non-trivial subgroups of D4. The second and third row consist of multicolored flowers representing the left (top row) and right (middle row) cosets of the subgroups on the top-most row. Each color in each flower is a coset. The bottom row of multicolored flowers represents the cosets of the subgroups on the bottom-most row. The bottom row of subgroups are normal, so their left and right cosets are the same. Please see the accompanying schematic.
