2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142393We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an FG-module, then the dimension of the second cohomology group of G with coefficients in M is at most 17.5 times the dimension of M. The third result is that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module. These last two results answer conjectures of Holt.44 pages, to appear in Groups, Geometry and DynamicsGroup TheoryRepresentation Theory20G10, 20D06text