1998 Fiscal Year Final Research Report Summary
Cohomology of Swan groups
| Project/Area Number |
09640058
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tohoku Institute of Technology |
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Principal Investigator |
OGAWA Yoshito Tohoku Institute of Technology, Faculty of Engineering, Associate Prof., 工学部, 助教授 (60160777)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1997 – 1998
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| Keywords | finite groups / cohomology / commutative rings |
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| Research Abstract |
We are interested in the commutative ring structure of cohomology rings of finite 2-groups with coefficients in a field of two elements To conjecture results and to verify them, we need computers. It is well-known 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 p-group, 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 mod-2 cohomology rings of finite abelian 2-groups by means of software Macaulay2 and Singular. This result can be proved by hand.
2. A finite p-group is called a Swan group, if the computation of cohomology rings for any finite groups with it as Sylow p-subgroups is reduced to that for normalizers of Sylow p-subgroups in the whole groups. Henn-Priddy state that almost all finite p-groups 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 p-compliments. The head investigator computed some cohowology rings for finite groups with Swan groups as Sylow p-subgroups by using a program finvar belong to the Singular package.
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