Is the group D₄ of symmetries of the square in Example 8.10 solvable?

Data from in Example 8.10

Let us form the dihedral group D₄ of permutations corresponding to the ways that two copies of a square with vertices 1, 2, 3, and 4 can be placed, one covering the other with vertices on top of vertices (see Fig. 8.11). D4 will then be the group of symmetries of the square. It is also called the octic group. Again, we choose seemingly arbitrary notation that we shall explain later. Naively, we are using p_i for rotations, µ_i for mirror images in perpendicular bisectors of sides, and δ_i for diagonal flips. There are eight permutations involved here. Let

Transcribed Image Text:

Po || P1 = P2 = P3 = 1 1 22 2 3 4 2 3 1 23 3 4 2 (4 4) 4 1). /1 2 3 4 3 4 1 2 1 2 3 4 1 2 3 M₁ = M₂= 81 = 8₂: = (²2 12 3 4 4 3 21 1 2 3 4 4 3 2 1 1 12 3 4) 3 4 2 1 4 2 4 3 3 4 3 2 3