Let D 4 be the group of symmetries of the square in Example 8 10 a Find the decomposition of D 4 into conjugate classes b Write the class equation for D 4 Data from Example 8 10 Let us form the dihedral group D 4 of permutations corresponding to the ways that two copies of a square with vertices 1,...

Let D 4 be the group of symmetries of the square in Example 8.10. a. Find the

Question:

Let D₄ be the group of symmetries of the square in Example 8.10.

a. Find the decomposition of D₄ into conjugate classes.

b. Write the class equation for D₄ .

Data from 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