When Do Random Subsets Decompose a Finite Group?

Abstract

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 union 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 \sqrt{Θ(G)|G|\log|G|}; i.e. for any c>0, if the size of A and B is less than (1-c)\sqrt{Θ(G)|G|\log|G|}, then with high probability (AB union BA) does not equal G. If A and B are larger than (1+c)\sqrt{Θ(G)|G|\log|G|} then (AB union BA) equals G with high probability.