Connectivity of the Product Replacement Graph of simple groups of bounded Lie rank

Abstract

The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating k-tuples of the group (for a fixed k). We show that there is a function c ( r ) such that for any finite simple group of Lie type, with Lie rank r, the Product Replacement Graph of the generating k-tuples is connected for any k ⩾ c ( r ) . The proof uses results of Larsen and Pink [M.J. Larsen, R. Pink, Finite subgroups of algebraic groups, preprint, 1998] and does not rely on the classification of finite simple groups.