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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Zero-one generation laws for finite simple groups
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by Robert M. Guralnick, Martin W. Liebeck, Frank Lübeck and Aner Shalev PDF
Proc. Amer. Math. Soc. 147 (2019), 2331-2347 Request permission

Abstract:

Let $G$ be a simple algebraic group over the algebraic closure of $\mathbb {F}_p$ ($p$ prime), and let $G(q)$ denote a corresponding finite group of Lie-type over $\mathbb {F}_q$, where $q$ is a power of $p$. Let $X$ be an irreducible subvariety of $G^r$ for some $r\ge 2$. We prove a zero-one law for the probability that $G(q)$ is generated by a random $r$-tuple in $X(q) = X\cap G(q)^r$: the limit of this probability as $q$ increases (through values of $q$ for which $X$ is stable under the Frobenius morphism defining $G(q)$) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs $G(q)$ to be generated by an $r$-tuple in $X(q)$ for two sufficiently large values of $q$. We also prove a version of this result where the underlying characteristic is allowed to vary.

In our main application, we apply these results to the case where $r=2$ and the irreducible subvariety $X = C\times D$, a product of two conjugacy classes of elements of finite order in $G$. This leads to new results on random $(2,3)$-generation of finite simple groups $G(q)$ of exceptional Lie-type: provided $G(q)$ is not a Suzuki group, we show that the probability that a random involution and a random element of order 3 generate $G(q)$ tends to $1$ as $q \rightarrow \infty$. Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and $\mathrm {PSp}_4(q)$) are randomly $(2,3)$-generated.

Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie-type.

References
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Additional Information
  • Robert M. Guralnick
  • Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
  • MR Author ID: 78455
  • Email: guralnic@usc.edu
  • Martin W. Liebeck
  • Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom
  • MR Author ID: 113845
  • ORCID: 0000-0002-3284-9899
  • Email: m.liebeck@imperial.ac.uk
  • Frank Lübeck
  • Affiliation: Lehrstuhl D für Mathematik, Pontdriesch 14/16, 52062 Aachen, Germany
  • MR Author ID: 362381
  • Email: frank.luebeck@math.rwth-aachen.de
  • Aner Shalev
  • Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
  • MR Author ID: 228986
  • ORCID: 0000-0001-9428-2958
  • Email: shalev@math.huji.ac.il
  • Received by editor(s): May 15, 2018
  • Received by editor(s) in revised form: July 2, 2018, and September 20, 2018
  • Published electronically: February 20, 2019
  • Additional Notes: The first author acknowledges the support of the NSF grant DMS-1600056.
    The third author was supported by project B3 of SFB-TRR 195 ‘Symbolic Tools in Mathematics and their Application’ of the German Research Foundation (DFG).
    The fourth author acknowledges the support of ISF grant 686/17 and the Vinik chair of mathematics which he holds. The authors also acknowledge the support of the National Science Foundation under Grant No. DMS-1440140 while they were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester.
  • Communicated by: Pham Huu Tiep
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 2331-2347
  • MSC (2010): Primary 20P05, 20G40, 20D06
  • DOI: https://doi.org/10.1090/proc/14404
  • MathSciNet review: 3951415