Presentations of finite simple groups: profinite and cohomological approaches

Robert M. Guralnick; William M. Kantor; Martin Kassabov; Alexander Lubotzky

doi:10.4171/ggd/22

JournalsggdVol. 1, No. 4pp. 469–523

Presentations of finite simple groups: profinite and cohomological approaches

Robert M. Guralnick
University of Southern California, Los Angeles, United States
William M. Kantor
University of Oregon, Eugene, United States
Martin Kassabov
Cornell University, Ithaca, United States
Alexander Lubotzky
Hebrew University, Jerusalem, Israel

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Abstract

We prove the following three closely related results:

Every finite simple group $G$ has a profinite presentation with 2 generators and at most 18 relations.
If $G$ is a finite simple group, $F$ a field and $M$ is an $FG$ -module, then $dim H^{2} (G, M) \leq (17.5) dim M$ .
If $G$ is a finite group, $F$ a field and $M$ is an irreducible faithful $FG$ -module, then $dim H^{2} (G, M) \leq (18.5) dim M$ .

Cite this article

Robert M. Guralnick, William M. Kantor, Martin Kassabov, Alexander Lubotzky, Presentations of finite simple groups: profinite and cohomological approaches. Groups Geom. Dyn. 1 (2007), no. 4, pp. 469–523

DOI 10.4171/GGD/22