Groups, Geometry, and Dynamics


Full-Text PDF (458 KB) | Metadata | Table of Contents | GGD summary
Volume 1, Issue 4, 2007, pp. 469–523
DOI: 10.4171/GGD/22

Published online: 2007-12-31

Presentations of finite simple groups: profinite and cohomological approaches

Robert M. Guralnick[1], William M. Kantor[2], Martin Kassabov[3] and Alexander Lubotzky[4]

(1) University of Southern California, Los Angeles, United States
(2) University of Oregon, Eugene, United States
(3) Cornell University, Ithaca, United States
(4) Hebrew University, Jerusalem, Israel

We prove the following three closely related results:

  1. Every finite simple group G has a profinite presentation with 2 generators and at most 18 relations.
  2. If G is a finite simple group, F a field and M is an FG-module, then  dim H2(G,M) ≤ (17.5) dim M.
  3. If G is a finite group, F a field and M is an irreducible faithful FG-module, then dim H2(G,M) ≤ (18.5) dim M.

Keywords: Finite simple groups, generators, relations, presentations, profinite presentations, cohomology, second cohomology group

Guralnick Robert, Kantor William, Kassabov Martin, Lubotzky Alexander: Presentations of finite simple groups: profinite and cohomological approaches. Groups Geom. Dyn. 1 (2007), 469-523. doi: 10.4171/GGD/22