Relative group cohomology and the orbit category

Abstract

Let G be a finite group and F be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative F -projective resolution for Z when F is the family of all subgroups H ≤ G with rk H ≤ rk G − 1. We answer this question negatively by calculating the relative group cohomology F H * (G, F 2 ) where G = Z/2 × Z/2 and F is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology F H * (G, M ) can be calculated using the ext-groups over the orbit category of G restricted to the family F . In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E 2 page is isomorphic to the relative group cohomology of G.