Small-diameter Cayley Graphs for Finite Simple Groups

Abstract

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 ❘.