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.
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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