Mixing and generation in simple groups

Aner Shalev

doi:10.1016/j.jalgebra.2007.07.031

Mixing and generation in simple groups

Aner Shalev^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

Let G be a finite simple group. We show that a random walk on G with respect to the conjugacy class x^G of a random element x ∈ G has mixing time 2. In particular it follows that (x^G)² covers almost all of G, which could be regarded as a probabilistic version of a longstanding conjecture of Thompson. We also show that if w is a non-trivial word, then almost every pair of values of w in G generates G.

Original language	English
Pages (from-to)	3075-3086
Number of pages	12
Journal	Journal of Algebra
Volume	319
Issue number	7
DOIs	https://doi.org/10.1016/j.jalgebra.2007.07.031
State	Published - 1 Apr 2008

Bibliographical note

Funding Information:
✩ Supported by the Israel Science Foundation and by the Bi-National Science Foundation United States–Israel Grant 2004-052. E-mail address: [email protected].

Keywords

Characters
Finite simple groups
Probabilistic methods
Random walks
Word maps

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10.1016/j.jalgebra.2007.07.031

Cite this

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N2 - Let G be a finite simple group. We show that a random walk on G with respect to the conjugacy class xG of a random element x ∈ G has mixing time 2. In particular it follows that (xG)2 covers almost all of G, which could be regarded as a probabilistic version of a longstanding conjecture of Thompson. We also show that if w is a non-trivial word, then almost every pair of values of w in G generates G.

AB - Let G be a finite simple group. We show that a random walk on G with respect to the conjugacy class xG of a random element x ∈ G has mixing time 2. In particular it follows that (xG)2 covers almost all of G, which could be regarded as a probabilistic version of a longstanding conjecture of Thompson. We also show that if w is a non-trivial word, then almost every pair of values of w in G generates G.

KW - Characters

KW - Finite simple groups

KW - Probabilistic methods

KW - Random walks

KW - Word maps

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JO - Journal of Algebra

JF - Journal of Algebra

IS - 7

ER -

Mixing and generation in simple groups

Abstract

Bibliographical note

Keywords

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