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Zero-one generation laws for finite simple groups

Authors: Robert M. Guralnick, Martin W. Liebeck, Frank Lübeck and Aner Shalev
Journal: Proc. Amer. Math. Soc. 147 (2019), 2331-2347
MSC (2010): Primary 20P05, 20G40, 20D06
Published electronically: February 20, 2019
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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

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom

Frank Lübeck
Affiliation: Lehrstuhl D für Mathematik, Pontdriesch 14/16, 52062 Aachen, Germany

Aner Shalev
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

Keywords: Generation of groups, random generation, asymptotic group theory, algebraic groups, generation
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
Article copyright: © Copyright 2019 American Mathematical Society