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Research on defect groups of blocks of a group ring

Research Project

Project/Area Number 10640004
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionGunma University

Principal Investigator

FUKUSHIMA Hiroshi  Gunma University, Faculty of Education, assistant professor, 教育学部, 助教授 (30125869)

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)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 1999: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1998: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordscharacter / finite group / representation / block / modular / defect / group ring / radical
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.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (41 results)

All Other

All Publications (41 results)

  • [Publications] Hiroshi Fukushima: "On the existence of p-blocks of defect O in p-nilpotent groups"J. Algebra. 222. 747-768 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Fukushima: "On the existence of p-blocks of defect O in solbable groups"J. Algebra. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Fukushima: "Group-theoretical characterization of p-radical groups"Sci. Rep. Fac. of Edu. Gunma Univ.. 47. 7-11 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Fukushima: "Some results on p-blocks of p-radical groups"Sci. Rep. Fac. of Edu. Gunma Univ.. 46. 9-13 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 大竹公一郎、武井修他: "簡約律と図形指導"群馬大学教育学部紀要. 48巻. (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 大竹公一郎、小柏香穂理: "ネットワークの構築について"群馬大学教育実践研究. 17号. (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takashi Itoh: "Completely positive decompositions from duals of C^*-algebras to von Neumann algebras"Math. Japonica. 51. 89-98 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takashi Itoh,Masaru Nagisa: "Schur products and module maps on B(H)"Research Inst. Math. Sci. Kyoto Univ.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 布川護,O.P.Ahuja: "Neighborhoods of analytic functions by Ruscheweyh deriuctives"Mathematica Japonica (accepted on July 16,1999). Vol.52(3). (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 布川護他3名: "On multivalently convex and starlike functions"Mathematica Japonica. Vol.49(2). 223-227 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.E.Cho,布川他2名: "Angular estimate of certain analytic functions"Indian Journal of Pure and Applied Mathematics. 29(3). 321-331 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Owa,布川護他2名: "Starlikeness of certain integrals"Mathematical Research Hot-Line. 2(1). 23-26 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 布川護: "Gamma-starlike関数について"京都大学数理解析研究所講究録. 1112. 44-50 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 布川護,O.P.Ahuja: "On meromorphic starlike and convex functions"京都大学数理解析研究所講究録. 1112. 51-56 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Fukushima: "On the existence of p-blocks of defect 0 in p-nilpotent groups"J. Algebra. 222. 747-768 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Fukushima: "On the existence of p-blocks of defect 0 in solvable groups"J. Algebra. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Fukushima: "Group-theoretical characterization of p-radical groups"Sci. Rep. Fac. of Edu. Gunma Univ.. 47. 7-11 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Fukushima: "Some results on p-blocks of p-radical groups"Sci. Rep. Fac. of Edu. Gunma Univ.. 46. 9-13 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takashi Itoh: "Completely positive decompositions from duals of CィイD1*ィエD1-algebras to von Neumann algebras"Math. Japonica. 51. 89-98 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takashi Itoh: "Schur products and module maps on B(H)"Research Inst. Math. Sci. Kyoto Univ.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nunokawa: "Neighborhoods of analytic functions by Ruscheweyh derivatives"Math. Japonica. 52)to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nunokawa: "On multivalently convex and starlike functions"Math. Japonica. 49. 223-227 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nunokawa: "Angular estimate of certain analytic function"Indian Journal of Pure and Applied Mathematics. 29. 321-331 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Fukushima: "On the existence of p-blocks of defect 0 in p-nilpotent groups"J. Algebra. 222. 747-768 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroshi Fukushima: "On the existence of p-blocks of defect 0 in solvable groups"J. Algebra. (掲載決定).

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroshi Fukushima: "Group-theoretical characterization of p-radical groups"Sci, Rep, Fac, of Edu, Gunma Univ.. 47. 7-11 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroshi Fukushima: "Some results on p-blocks of p-radical groups"Sci, Rep, Fac, of Edu, Gunma Univ.. 46. 9-13 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] 大竹公一郎,武井修他: "簡約律と図形指導"群馬大学教育学部紀要. 48巻. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 大竹公一郎,武井修他: "ネットワークの構築について"群馬大学教育実践研究. 17号. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Takashi Itoh: "Completely positive decompositions from duals of C^*-algebrus to von Neumann algebras"Math. Japonica. 51. 89-98 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Takashi Itoh , Masaru Nagisa: "Schur products and module maps on B(H)"Research Inst. Math.Sci, Kyoto univ.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] 布川護,O.P.Ahuja: "Neighborhoods of analytic fcnctions by Ruscheweyh derivatives"Mathematica Japonica (accepted on July 16,1999). vol.52(3). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 布川護 他3名: "On multivalently convex and starlike functions"Mathematica Japonica. vol.49(2). 223-227 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] N.E.Cho. 布川他2名: "Augular estimate of certain analytic functions"Indian Journal of Pure and Applied Mathematics. 29(3). 321-331 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Owa,布川護他2名: "Starlikeness of certain integrals"Mathematical Research Hot-Line. 2(1). 23-26 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] 布川護: "Gamma-starlike関数について"京都大学数理解析研究所講究録. 1112. 44-50 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 布川護,O.P.Ahuja: "On meromorphic storlik and convex functioris"京都大学数理解析研究所講究録. 1112. 51-56 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroshi Fukushima: "Group-theoretical characterization of P-radical groups" Science Reports of Education,Gunma University. 47. 7-11 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiroshi Fukushima: "Some results on P-blocks of P-radical groups" Science Reports of Education,Gunma Uniuersity. 46. 9-13 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiroshi Fukushima: "On the existence of P-blocks of defect 0" Science Reports of Education,Gunma Unuiersity. 45. 9-15 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 二宮 晏: "線形代数入門" 学術図書出版, 59 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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