When Do Random Subsets Decompose a Finite Group?

TL;DR

This paper investigates the conditions under which two random subsets of a finite group generate the entire group through their products, revealing a phase transition related to the subsets' sizes and a new group invariant.

Contribution

It introduces the invariant (G) and characterizes a phase transition threshold for random subsets to decompose a finite group.

Findings

(G) ranges between 1/2 and 1, equaling 1 iff the group is abelian.

A phase transition occurs at subset sizes around (G)|G| log|G|.

Below the threshold, the product set likely does not cover G; above it, it does.

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 \Theta(G). \Theta(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{\Theta(G)|G|\log|G|}; i.e. for any c>0, if the size of A and B is less than (1-c)\sqrt{\Theta(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{\Theta(G)|G|\log|G|} then (AB union BA) equals G with high probability.