When do random subsets decompose a finite group?

Let A, B be two random subsets of a finite group G . We consider the event that the products of elements from A and B span the whole group, i.e. [ AB ∪ BA = G ]. The study of this event gives rise to a group invariant we call Θ( G ). Θ( G ) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes √Θ( G )| G | log | G |; i.e. for any ɛ > 0, if the size of A and B is less than (1 − ɛ )√Θ( G )| G | log | G |, then with high probability AB ∪ BA ≠ G . If A and B are larger than (1 + ɛ )√Θ( G )| G | log | G |, then AB ∪ BA = G with high probability.