Let X be the D 4 set in Example 16 8 a Does D 4 act faithfully on X b Find all orbits in X on which D 4 acts faithfully as a sub D 4 set Data from in Example 16 8 Let G be the group D 4 p 0 , p 1 , p 2 , p 3 , 1 , 2, 1 , 2 of symmetries of the square, described In Fig 16 9 we show the square with v...

Let X be the D 4 -set in Example 16.8. a. Does D 4 act faithfully on

Question:

Let X be the D₄-set in Example 16.8.

a. Does D₄ act faithfully on X?

b. Find all orbits in X on which D₄ acts faithfully as a sub-D₄-set.

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