Imperial College London
Professor Martin Liebeck

ProfessorMartinLiebeck

Faculty of Natural Sciences, Department of Mathematics

Head of Pure Mathematics Section/Prof of Pure Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 8490m.liebeck Website

 
 
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Location

 

665Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Liebeck:2019:proc/14404,
author = {Liebeck, MW and Guralnick, R and Shalev, A and Lubeck, F},
doi = {proc/14404},
journal = {Proceedings of the American Mathematical Society},
pages = {2331--2347},
title = {Zero-one generation laws for finite simple groups},
url = {http://dx.doi.org/10.1090/proc/14404},
volume = {147},
year = {2019}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - Let G be a simple algebraic group over the algebraic closure of Fp (p prime), and let G (q) denote a corresponding finite group of Lie type over Fq, where q is a power of p.  Let X be an irreducible subvariety of Gr for some r≥2. We prove a zero-one law for the probability that G(q) is generated by a random r-tuple in X(q) =X ∩ 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 × 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→∞. Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and PSp4(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.
AU  - Liebeck,MW
AU  - Guralnick,R
AU  - Shalev,A
AU  - Lubeck,F
DO  - proc/14404
EP  - 2347
PY  - 2019///
SN  - 0002-9939
SP  - 2331
TI  - Zero-one generation laws for finite simple groups
T2  - Proceedings of the American Mathematical Society
UR  - http://dx.doi.org/10.1090/proc/14404
UR  - http://hdl.handle.net/10044/1/65029
VL  - 147
ER  -
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