Show how all possible G-sets, up to isomorphism (see Exercise 9), can be formed from the group G.

Let X_i for i ∈ I be G-sets for the same group G, and suppose the sets X_i are not necessarily disjoint. Let x'_i = {(x, i) |x ∈ X_i} for each i ∈ I. Then the sets X'_i are disjoint, and each can still be regarded as a G-setin an obvious way. (The elements of X_i have simply been tagged by i to distinguish them from the elements of X_j for i ≠ j.) The G-set U_i∈I  X'_i is the disjoint union of the G-sets X;. Using Exercises 14 and 15, show that every G-set is isomorphic to a disjoint union of left coset G-sets, as described in Example 16.7.

Data from in Example 16.7

Let H be a subgroup of G, and let L_H be the set of all left cosets of H. Then L_H is a G-set, where the action of g ∈ G on the left coset xH is given by g(xH) = (gx)H. Observe that this action is well defined: if yH = xH, then y = x h for some h ∈ H, and g(yH) = (gy)H = (gxh)H = (gx)(hH) = (gx)H = g(xH). A series of exercises shows that every G-set is isomorphic to one that may be formed using these left coset G-sets as building blocks.

Data from Exercise 9

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?