Exercises through show how all possible G-sets, up to isomorphism (see Exercise 9), can be formed from the group G. Let Xi, for i ∈ I be G-sets for the same group G, and suppose the sets Xi, are not necessarily disjoint. Let Xi’ = {(x, i)ǀ X ∈ Xi} for each i ∈ I. Then the sets Xi’ are disjoint, and each can still be regarded as a G-set in an obvious way. (The elements of X, have simply been tagged by 1 to distinguish them from the elements of Xj for i ≠ j.) The G-set ∪i∈I,Xi 'is me disjoint union of the G-sets Xi. 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.
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