Covering Numbers for Simple Algebraic Groups

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)≤cdimGdimC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$cn(G,C) \le c \frac {\dim G}{\dim C}$\end{document}, where c is an explicit constant (at most 120). Some consequences on the width and generation of simple algebraic groups are given.