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 H2(G,M) ≤ (17.5) dim M.
- 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.
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. 469–523DOI 10.4171/GGD/22