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

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

2007-10-02

arxiv.org/abs/0710.0437v2

J. Algebra 320 (2008), 945-960

Mathematics

Group Theory

v2 Minor changes, some explanations added

Scientific paper

10.1016/j.jalgebra.2008.03.005

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.

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