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

Nir Avni*, Shelly Garion

*Corresponding author for this work

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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.

Original language English (US) 945-960 16 Journal of Algebra 320 2 https://doi.org/10.1016/j.jalgebra.2008.03.005 Published - Jul 15 2008

### Fingerprint

Simple group
Replacement
Connectivity
Finite Simple Group
Graph in graph theory
Groups of Lie Type
Random Element
Algebraic Groups
Random walk
Finite Group
Subgroup

### Keywords

• Product Replacement Algorithm
• T-systems

### ASJC Scopus subject areas

• Algebra and Number Theory

### Cite this

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title = "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.",
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In: Journal of Algebra, Vol. 320, No. 2, 15.07.2008, p. 945-960.

Research output: Contribution to journalArticle

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T1 - Connectivity of the Product Replacement Graph of simple groups of bounded Lie rank

AU - Avni, Nir

AU - Garion, Shelly

PY - 2008/7/15

Y1 - 2008/7/15

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AB - 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.

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