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

Let X be a transitive G-set, and let x₀ ∈ X. Show that X is isomorphic (see Exercise 9) to the G-set L of all left cosets of G_x0 , 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.