Cohmology theory of finite groups

我们证明了一些对有限群上同调代数计算有用的基本定理。即证明了关于Carlson模的顶点的一个事实，射影覆盖相对于模与Green对应的关系，Carlson模与Green对应的关系，以及当p-秩为2时由一个生产类构造的参数系统。其中p是素数.我们构造了一个模，它出现在有限群G的上同调代数的研究中，使得(0)P有秩2(1)P的中心是循环的(2)所有二阶初等交换子群的中心化子都是G的中心化子中的交换子和Sylow子群，从而可以统一地研究这些有限群的上同调代数.作为应用，我们首先计算了有限群的有圈Sylow 2-子群的mod2上同调代数.其次，我们研究了具有p^3阶超特殊Sylow p-子群和指数p的有限群的mod p上同调代数，建立了研究的一般框架。作为例子，我们计算了一般线性群GL(3，F_p)，p&gt；3的上同调代数。在这类有限群中，已知mod p上同调代数的群只有Mathieu群M_&lt；12&gt；和GL(3，F_3)(这两个有限群有同构的上同调代数)。我们将能够计算具有相同类型的Sylow p-子群的零星有限单群的上同调代数。

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.