I describe recent work with Pyber, Short and Szabo in which we study the `width' of a finite simple group. Given a group G and a subset A of G, the `width of G with respect to A' - w(G,A) - is the smallest number k such that G can be written as the product of k conjugates of A. If G is finite and simple, and A is a set of size at least 2, then w(G,A) is well-defined; what is more Liebeck, Nikolov and Shalev have conjectured that in this situation there exists an absolute constant c such that w(G,A)\leq c log|G|/log|A|.
I will present a partial proof of this conjecture as well as describing some interesting, and unexpected, connections between this work and classical additive combinatorics. In particular the notion of a normal K-approximate group will be introduced.
- Algebra Seminar