Exercises through 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 H and 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 g0xo and if Gx0 ≠ Gg0x0, then the collections LH of left cosets of H = Gx0 and Lk of left cosets of K = Gg0x0 form distinct G-sets that must be isomorphic, since both LH and Lk are isomorphic to X.
a. Let X be a transitive G-set and let X0 ∈ X and g0 ∈ G. If H = Gx0, describe K = Gg0x0 in terms of H and g0.
b. Based on part (a), conjecture conditions on subgroups H and K of G such that the left coset G-sets of H and K are isomoiphic.
c. Prove your conjecture in part (b).
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