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

The preceding exercises show that every G-set X is isomorphic to a disjoint union of left coset G-sets. The question then arises whether left coset G-sets of distinct subgroups Hand K of G can themselves be isomorphic. Note that the map defined in the hint of Exercise 15 depends on the choice of x₀ as "base point." If x₀ is replaced by g₀x₀ and if G_x0 ≠ Gg₀x₀ , then the collections L_H of left cosets of H = G_xo and L_K ofleft cosets of K = G_g0x0 form distinct G-sets that must be isomorphic, since both L_H and L_K are isomorphic to X. 

a. Let X be a transitive G-set and let x₀ ∈ X and g₀ ∈ G. If H = G_x0 describe K = G_g0x0 in terms of H and g₀. 

b. Based on part (a), conjecture conditions on subgroups Hand K of G such that the left coset G-sets of H and K are isomorphic. 

c. Prove your conjecture in part (b ).

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.