Cohomology of infinite groups realizing fusion systems

Abstract

Given a fusion system FF defined on a p-group S, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize FF. We study these models when FF is a fusion system of a finite group G and prove a theorem which relates the cohomology of an infinite group model ππ to the cohomology of the group G. We show that for the groups GL(n, 2), where n≥5n≥5, the cohomology of the infinite group obtained using the Robinson model is different than the cohomology of the fusion system. We also discuss the signalizer functors P→Θ(P)P→Θ(P) for infinite group models and obtain a long exact sequence for calculating the cohomology of a centric linking system with twisted coefficients.