Let X and Y be G-sets with the same group G. An isomorphism between G-sets X and Y is a map ∅ : X → Y that is one to one, onto Y, and satisfies g∅(x) = ∅(gx) for all x ∈ X and g ∈ G. Two G-sets are isomorphic if such an isomorphism between them exists. Let X be the D₄-set of Example 16.8.

a. Find two distinct orbits of X that are isomorphic sub-D₄-sets. 

b. Show that the orbits {l, 2, 3, 4} and {s₁, s₂ , s₃ , s₄} are not isomorphic sub-D4-sets.

c. Are the orbits you gave for your answer to part (a) the only two different isomorphic sub-D₄-sets of X?  

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