1999 Fiscal Year Final Research Report Summary
Research on defect groups of blocks of a group ring
| Project/Area Number |
10640004
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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 | Gunma University |
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Principal Investigator |
FUKUSHIMA Hiroshi Gunma University, Faculty of Education, assistant professor, 教育学部, 助教授 (30125869)
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| Co-Investigator(Kenkyū-buntansha) |
ITOH Takashi Gunma University, Faculty of Education, assistant professor, 教育学部, 助教授 (40193495)
NINOMIYA Yasushi Shinsyu University, Faculty of Science, professor, 理学部, 教授 (40092887)
OTAKE Koichiro Gunma University, Faculty of Education, professor, 教育学部, 教授 (60134269)
NUNOKAWA Mamoru Gunma University, Faculty of Education, professor, 教育学部, 教授 (00008137)
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| Project Period (FY) |
1998 – 1999
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| Keywords | character / finite group / representation / block / modular / defect / group ring / radical |
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| Research Abstract |
Blocks of defect zero are easily understood objects in modular representation theory, since they are isomorphic to full matrix rings over the underlying field. Brauer's "Problem 19" asks for group-theoretic criteria to determine when a group has such blocks. In connection with this problem, we proved the following result : let G be a solvable p-nilpotent group for some prime p, such that OィイD2pィエD2(G) = 1. If certain groups is not involved in G, an element in OィイD2p'ィエD2(G) with a centraliser of p'-order is found. Hence G has a p-block of defect zero.
Next we proved the following result : Let G be a finite solvable group and p, a prime. Then G has a character of p-defect 0 if and only if G has a chain of subgroups G ⊇ GィイD21ィエD2 ⊇ GィイD22ィエD2 ⊇ … ⊇ GィイD2nィエD2, which satisfies the some conditions. By using the theorem, we can investigate without charactertable whether given groups have blocks of defect zero.
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Research Products
(23 results)
