Connectivity of the Product Replacement Algorithm Graph of PSL(2,q)

TL;DR

This paper proves that the graph of generating k-tuples for PSL(2,q) and PGL(2,q) is connected for all k≥4, extending previous results and ensuring the effectiveness of the product replacement algorithm for these groups.

Contribution

It establishes the connectivity of the product replacement graph for PSL(2,q) and PGL(2,q) for all k≥4, generalizing earlier findings.

Findings

The graph is connected for PSL(2,q) and PGL(2,q) when k≥4.

Connectivity holds for all prime powers q.

Extends previous results by Gilman and Evans.

Abstract

The product replacement algorithm is a practical algorithm to construct random elements of a finite group G. It can be described as a random walk on a graph whose vertices are the generating k-tuples of G (for a fixed k). We show that if G is PSL(2,q) or PGL(2,q), where q is a prime power, then this graph is connected for any k>=4. This generalizes former results obtained by Gilman and Evans.