Zur Kohomologie von Gruppen mit abelschen Sylowuntergruppen

Abstract

In 1962 Atiyah showed, with, the help of topological methods, the existence of a spectral sequence joining the cohomology ring of a finite group (with values in the rational integers) and the ring of generalized complex characters of the group. This paper undertakes to get that connection (at least in part) in a purely algebraic way. To that effect such ring filtrations of the character ring of a group are described that are functorially compatible with group morphisms and compatible with the induction of representations belonging to subgroups. The graded ring formed canonically by a filtration of that kind is to be compared with the cohomology ring of the same group, and it is shown, the group assumed to have abelian Sylow subgroups or to be a periodic group in a cohomological sense, that by taking a certain filtration its graded ring can be identified with a homomorphic part of the cohomology ring. (That corresponds to the infinite term of Atiyah's spectral sequence). The generalization of this result with respect to all finite groups has not yet been achieved; there is, however, a reduction theorem saying that it is sufficient to deal with groups of prime power order only. - As an application one gets lower bounds for the orders of the cohomology groups belonging to a finite group, especially, as a corollary, the theorem of its infinite cohomological dimension.