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

Let {X_i |i ∈ I} be a disjoint collection of sets, so X_i ∩ X_j = ∅ for i ≠ j. Let each X_i be a G-set for the same group G. 

a. Show that U_iEIX_i can be viewed in a natural way as a G-set, the union of the G-sets X_i. 

b. Show that every G-set X is the union of its orbits.  

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.