let X={l, 2, 3, 4, s₁.s₂,s₃,s₄,m₁,m₂,d₁,d₂,C,P₁,P₂,P₃,P₄} be the D₄-set of Example 16.8 with action table in Table 16.10. Find the following, where G = D₄.

The orbits in X under D₄

Data from in Example 16.8

Let G be the group D₄ = {p₀, p₁, p₂, p₃, μ₁, μ2, δ₁, δ₂ } of symmetries of the square, described. In Fig. 16.9 we show the square with vertices 1, 2, 3, 4 as in Fig. 8.11. We also label the sides s₁ s₂, s₃ , s₄ , the diagonals d₁ and d₂, vertical and horizontal axes m₁ and m₂, the center point C, and midpoints P_i of the sides s_i. Recall that p_i corresponds to rotating the square counterclockwise through πi/2 radians, µ_i

Corresponds to flipping on the axis m_i and δ_i to flipping on the diagonal d_i. We let

X = {1, 2, 3, 4, s₁, s₂, s₃, s₄, m₁, m₂, d₁, d₂, C, P₁, P₂, P₃, P₄}. 

Then X can be regarded as a D₄-set in a natural way. Table 16.10 describes completely the action of D₄ on X and is given to provide geometric illustrations of ideas to be introduced. We should be sure that we understand how this table is formed before continuing