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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Presentations of finite simple groups: A quantitative approach


Authors: R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky
Journal: J. Amer. Math. Soc. 21 (2008), 711-774
MSC (2000): Primary 20D06, 20F05; Secondary 20J06
Published electronically: February 18, 2008
MathSciNet review: 2393425
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Abstract: There is a constant $ C_0$ such that all nonabelian finite simple groups of rank $ n$ over $ \mathbb{F}_q$, with the possible exception of the Ree groups $ ^2G_2(3^{2e+1})$, have presentations with at most $ C_0$ generators and relations and total length at most $ C_0(\log n +\log q)$. As a corollary, we deduce a conjecture of Holt: there is a constant $ C$ such that $ \dim H^2(G,M) \leq C\dim M$ for every finite simple group $ G$, every prime $ p$ and every irreducible $ {\mathbb{F}}_p G $-module $ M$.


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Additional Information

R. M. Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: guralnic@usc.edu

W. M. Kantor
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: kantor@math.uoregon.edu

M. Kassabov
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
Email: kassabov@math.cornell.edu

A. Lubotzky
Affiliation: Department of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel
Email: alexlub@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0894-0347-08-00590-0
PII: S 0894-0347(08)00590-0
Received by editor(s): February 22, 2006
Published electronically: February 18, 2008
Additional Notes: The authors were partially supported by NSF grants DMS 0140578, DMS 0242983, DMS 0600244 and DMS 0354731. The authors are grateful for the support and hospitality of the Institute for Advanced Study, where this research was carried out. The research by the last author was also supported by the Ambrose Monell Foundation and the Ellentuck Fund.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.