Connectivity of the Product Replacement Graph of Simple Groups of Bounded Lie Rank

TL;DR

This paper proves that for finite simple groups of Lie type with bounded Lie rank, the product replacement graph of generating k-tuples is connected for sufficiently large k, independent of the classification of finite simple groups.

Contribution

It establishes connectivity of the product replacement graph for simple groups of Lie type with bounded rank, using new methods that avoid reliance on the classification theorem.

Findings

Connectivity holds for k > c(r) for simple groups of Lie type

The proof uses results of Larsen and Pink

No classification of finite simple groups needed

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 integer 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 and does not rely on the classification of finite simple groups.