Covering Numbers for Simple Algebraic Groups

Abstract

Let G be a simple algebraic group over an algebraically closed field, and let C be a noncentral conjugacy class of G. The covering number cn(G,C) is defined to be the minimal k such that G = Ck, where Ck = {c1c2⋯ck : ci ∈ C}. We prove that cn(G,C) le c frac {dim G}{dim C}, where c is an explicit constant (at most 120). Some consequences on the width and generation of simple algebraic groups are given.