Cohomology of Swan groups
Project/Area Number  09640058 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Tohoku Institute of Technology 
Principal Investigator 
OGAWA Yoshito Tohoku Institute of Technology, Faculty of Engineering, Associate Prof., 工学部, 助教授 (60160777)

CoInvestigator(Kenkyūbuntansha) 
KURODA Tadashi Tohoku Institute of Technology, Faculty of Engineering, Prof., 工学部, 教授 (40004238)
SATO Kojro Tohoku Institute of Technology, Faculty of Engineering, Prof., 工学部, 教授 (10085491)

Project Period (FY) 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1997 : ¥700,000 (Direct Cost : ¥700,000)

Keywords  finite groups / cohomology / commutative rings 
Research Abstract 
We are interested in the commutative ring structure of cohomology rings of finite 2groups with coefficients in a field of two elements To conjecture results and to verify them, we need computers. It is wellknown that Carlson investigates cohomology rings of finite groups by virtue of software MAGMA (http : //www. math. uga. edu/ifc). 1. In a cohomology ring of a finite pgroup, an essential ideal is the ideal whose elements cannot be detected by using any family of proper subgroups. In 1982, Muiconjectured that the square of an essential ideal is zero. This conjecture is yet to be solved. The head investigator computed essential ideals and their shortest primary decompositions for mod2 cohomology rings of finite abelian 2groups by means of software Macaulay2 and Singular. This result can be proved by hand. 2. A finite pgroup is called a Swan group, if the computation of cohomology rings for any finite groups with it as Sylow psubgroups is reduced to that for normalizers of Sylow psubgroups in the whole groups. HennPriddy state that almost all finite pgroups are Swan groups in some sense ; nevertheless the classification of Swan groups is very difficult. Moreover, the computation of the cohomology rings for normalizers above is reduced to that of invariant subrings of the cohomology ring of the Swan group by Sylow pcompliments. The head investigator computed some cohowology rings for finite groups with Swan groups as Sylow psubgroups by using a program finvar belong to the Singular package.

Report
(3results)
Research Products
(9results)