Divisibility and laws in finite simple groups

We provide new bounds for the divisibility function of the free group F2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf F}_2$$\end{document} and construct short laws for the symmetric groups Sym(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{Sym}}}(n)$$\end{document}. The construction is random and relies on the classification of the finite simple groups. We also give bounds on the length of laws for finite simple groups of Lie type.