Cohomology theory of finite groups

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As a continuation of our research on mod p cohomology algebras of finite groups with extraspecial Sylow p-subgroups, which was done under Grant-in-Aid for Scientific Research during 1997〜1998 (project number 096400460), we calculated the mod 7 cohomolgy algebra of Held simple group. In this work we completed the theoreical frame work on the cohomology algebras of finite groups of this kind. We also calculated the mod p cohomology algebra of the special linear group of degree 3 over the prime field of characteristic p.We improved a theorem of Carlson on system of parameters. Namely if a finite group G has p-rank r, then the mod p cohomology algebra has a system of parameters ζ_1,...,ζ_r with the following properties: (1) for each i = 1,...,r, the element ζ_i is a sum of transfers from the centralizers of elementary abelian p=subgrpups of rank i; (2) for each i = 1,...,r, the restriction of {ζ_1,...,ζ_i} to an elementary abelian p-subgroup of rank i is a system of paramters of the cohomology algebra of this elementary abelian p-subgroup. From this fact we, in particular, showed that if a finite group G has p-rank less than or equal to 3, then the trivival kG-module k has index zero.Using transfer maps of extension groups introduced by Carlson, Peng, Wheeler, we showed that an element ρ in the mod p cohomology algebra of a finite group G is regular if and only if the transfer map Tr^<L_p> : Ext^*_<kG>(L_ρ,L_ρ) → Ext^*_<kG>, (k,k) defined by the Carlson module L_ρ is the zero map. Relating to this result we proved that if a finitely generated kG-module W is protective over a cyclic shifted subgroup in the center of a Sylow p-subgroup, then the transfer map Tr^W : Ext^*_<kG>(W,W) → Ext^*_<kG>(k,k) defined by the module W is the zero map.