Cohmology theory of finite groups

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We proved some fundamental theorems which are useful to calculation of cohomology algebras of finite groups. Namely we showed a fact on vertices of Carlson modules, relations of projective covers relative to modules and Green correspondence, relations of Carlson modules and Green correspondence, and a construction of system of parameters from a productive class when p-rank is two, where p is a prime number.We constructed a module which appears in investigation of cohomology algebra of finite group G with Sylow p-subgroup P such that (0) P has rank two (1) the center of P is cyclic (2) the centralizers of all elementary abelian subgroups of rank two are abelian and Sylow subgroups in the centralizers in G.Thereby the cohomology algebras of these finite groups can be investigated in a uniform way.As applications, first we calculated the mod 2 cohomology algebras of finite groups with wreathed Sylow 2-subgroups. Second we studied mod p cohomology algebras of finite groups with extraspecial Sylow p-subgroups of order p^3 and exponent p ; we constructed a general framework for the investigation. As an example we calculated the cohomology algebra of the general linear group GL(3, F_p), p > 3. Among such finite groups, the groups whose mod p cohomology algebras have been known are only Mathieu group M_<12> and GL(3, F_3) (these two finite groups have isomorphic cohomology algebras). We would be able to calculate cohomology algebras of sporadic finite simple groups with the same kind of Sylow p-subgroups.