Small-diameter Cayley Graphs for Finite Simple Groups

Let S be a subset generating a finite group G. The corresponding Cayley graph G(G, S) has the elements of G as vertices and the pairs {g, sg}, g ∈ G, s ∈ S, as edges. The diameter of G(G, S) is the smallest integer d such that every element of G can be expressed as a word of length ⩿d using elements from S ∪ S⁻¹. A simple count of words shows that d ⩾ log_{2 ❘s❘} (❘G❘). We prove that there is a constant C such that every nonabelian finite simple group has a set S of at most 7 generators for which the diameter of G(G, S) is at most C log ❘G❘.