COMMUNICATIONS IN ALGEBRA, vol.42, no.7, pp.3220-3243, 2014 (Journal Indexed in SCI)
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 HG with rkHrkG-1. We answer this question negatively by calculating the relative group cohomology FH*(G, ?(2)) where G = Z/2xZ/2 and F is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology FH*(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.