Subgroup Complexes

Abstract

In the computer calculations to compute the mod-p cohomology ring H*(G, k) of a finite group G, we first calculate the cohomology ring of the Sylow p-subgroup S of G. If G is not a p-group, extracting the cohomology of G as a subring of the cohomology ring of S is often a matter of finding the invariant elements. This reduces to an application of some sort of invariant theory. There must be some way of determining the action of the group on its p-subgroups. However, in some cases it is possible to construct the cohomology from the cohomology of proper subgroups of the group G. One such method uses subgroup complexes which are topological spaces constructed from partially ordered sets of collections of p-subgroups of G. We give an account of the method in this chapter.