Relative projectivity of modules and cohomology theory of finite groups (Q5960127)

The principal aim of this paper is to calculate the mod \(2\) cohomology algebras of finite groups with a wreathed Sylow \(2\)-subgroup, denoted by \(S\). Such groups are classified into four types according to the properties of a complete set of representatives of conjugacy classes of four-groups in \(S\), and each type is thoroughly examined. The authors use techniques from group cohomology developed especially by J.~F.~Carlson, and additional general results are proved in the first part of the paper. The authors investigate indecomposable direct summands of Carlson modules of homogeneous cohomology elements, and prove a theorem relating the projective covers of modules relative to modules and Green correspondence. The relative projectivity of Carlson modules is also studied.