Let G be a finite simple group and S a normal subset of G. If |G| is large enough in terms of |S|/|G|, can we deduce that every element of G can be expressed as x y-1 for x and y elements of S? Shalev, Tiep, and I have proven that this is true assuming G is an alternating group or a group of Lie type in bounded rank, but the question remains open for classical groups of high rank over small fields. I will say something about the methods of proof, which involve both character methods and geometric ideas and also say something about the more general question of covering G by ST where S and T are large normal subsets.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
