The geometry and cohomology of the Mathieu group M12

Abstract

It is usually very complex to calculate the cohomology of a finite group G. To date the most significant results have been on the symmetric and alternating groups ([N], [M– M], [Ma], [Mu], [A–M–M]); the general linear groups over a finite field [Q1]; and the extra–special p–groups [Q2]. In the list above only the alternating groups and the GLn(F2) are simple. Indeed, among the simple groups only a very few, aside from those above have been understood. It is probable that away from the characteristic, p, H(PSLn(Fpn );Z/q) is available, but even this is not obvious, especially for q = 2. And certainly we currently have no information on most of the families of groups of Lie type, for example, G2(Fq ), E6(Fq ), E7(Fq ), E8(Fq ), or any of the sporadic groups but M11 and J1. About 15 years ago Quillen introduced very powerful techniques that allowed for a much deeper understanding of group cohomology. However, it is also apparent that successful calculations over Z/p have only occurred when the group has a well–behaved lattice of p–elementary abelian subgroups. Formidable combinatorial problems arise in the general situation, as well as necessarily complicated multiplicative relations. For example, the general calculation of H(GLn(Fp);Z/p) remains quite inaccessible, and there has been only marginal progress since Quillen’s landmark results.