This permutation group is known, as an abstract group, as the dihedral group of order 8. The only remaining symmetry is the identity (1)(2)(3)(4). The reflection about the 1,3−diagonal line is (24) and reflection about the 2,4−diagonal is (13). The reflection about the horizontal line through the center is given by (12)(34) and the corresponding vertical line reflection is (14)(23). The rotation by 90° (counterclockwise) about the center of the square is described by the permutation (1234). The symmetries are determined by the images of the vertices, that can, in turn, be described by permutations. Let the vertices of a square be labeled 1, 2, 3 and 4 (counterclockwise around the square starting with 1 in the top left corner). This permutation group is, as an abstract group, the Klein group V 4.Īs another example consider the group of symmetries of a square. G 1 forms a group, since aa = bb = e, ba = ab, and abab = e.
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