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Classification of ideals in integral group rings

Research Project

Project/Area Number 03640041
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionAichi University of Education

Principal Investigator

TAHARA Ken-ichi  Aichi Univ.of Educ.Math.Sci.Professor, 教育学部・総合科学課程, 教授 (00024026)

Co-Investigator(Kenkyū-buntansha) YASUMOTO Taichi  Aichi Univ.of Educ.Math.Sci.Assistant, 教育学部・総合科学課程, 助手 (00231647)
SASAKI Moritoshi  Aichi Univ.of Educ.Math.Sci.Assist.Professor, 教育学部・総合科学課程, 助教授 (90178666)
TAKEUCHI Yoshihiro  Aichi Univ.of Educ.Math.Sci.Assist.Professor, 教育学部・総合科学課程, 助教授 (10206956)
FURUKAWA Yasuyuki  Aichi Univ.of Educ.Math.Sci.Professor, 教育学部・総合科学課程, 教授 (90024033)
HAYASHI Makoto  Aichi Univ.of Educ.Math.Sci.Professor, 教育学部・総合科学課程, 教授 (40109369)
Project Period (FY) 1991 – 1993
Project Status Completed (Fiscal Year 1993)
Budget Amount *help
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1993: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1992: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1991: ¥700,000 (Direct Cost: ¥700,000)
KeywordsAugmentation ideal / Lie power / Lie dimension subgroup / 添加イテアル / リー次元部分群 / リー次元部分群問題 / リ-績 / リ-次元部分群
Research Abstract

Let G be a group with the lower central series G = G_1 G_2 ・・・ G_n G_<n+1> ・・・. Denote by DELTA(G) the augmentation ideal of the integral group ring ZG of G over Z, the ring of rational integers. We define inductively the Lie powers DELTA^<(n)> (G) of DELTA(G) : DELTA^<(1)>(G)= DELTA(G), DELTA^<(n)>(G)=(DELTA^<(N-1)>(G), DELTA(G))ZG.Here (DELTA^<(n-1)>(G), DELTA(G)) is generated by (a, b)= ab - ba, aepsilonDELTA^<(n-1)>(G), bepsilonDELTA(G). Then we define nth Lie dimension subgroup D_<(n)>(G) of G as follows ;
D_<(n)>(G)=G (1+DELTA^<(n)>(G)).
R.Sandling proved D_<(n)>(G)= G_n for any group G and any n with 1(〕SY.ltoreq.〔)n(〕SY.ltoreq.〔)6 in 1972, and recently T.C.Hurley and S.K.Sehgal constracted groups G such that D_<(n)>(G) * G_n for any n(〕SY.gtoreq.〔)9.
We proved the exponent of D_<(n)>(G)/G_n devides 2 for any group G in 1991, and hence D_<(7)>(G) = G_7 for any p-group G with odd prime p. Furthermore, in 1992-93, we proved that D_<(n)>(G)= G_n for any group G and n =7, 8.
Therefore, the Lie dimension subgroup problem is solved completely, namely, D_<(n)>(G)=G_n for any group G and any n with (〕SY.ltoreq.〔)n(〕SY.ltoreq.〔)8, and there exist groups G such that D_<(n)>(G)*G_n for any n(〕SY.gtoreq.〔)9.

Report

(4 results)
  • 1993 Annual Research Report   Final Research Report Summary
  • 1992 Annual Research Report
  • 1991 Annual Research Report
  • Research Products

    (21 results)

All Other

All Publications (21 results)

  • [Publications] K.Gomi and M.Hayashi: "A pushing-up approach to the quasithin simple jinite groups with solvable 2-local subgroups" J.Algebra. 146. 412-426 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Hayashi and Y.Tanaka: "Amalgams of solvable groups" J.Fac.Sci.Univ.of Tokyo Sec.IA. 39. 309-338 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Narain Gupta and Ken-Ichi Tahara: "The seventh and eighth Lie dimension subgroups" J.Pure and Appl.Algebra. 88. 107-117 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Ken-Ichi Tahara and Jiam Xiao: "On the seventh dimension subgroups" Japan.J.Math.New Ser.20(予定). (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Ken-Ichi Tahara and Jiam Xiao: "Supplements on our paper“On the seventh dimension subgroups"" Bull.Aichi Univ.of Educ.43(予定). (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] K.Gomi and M.Hayashi: "A pushing-up approach to the quasithin simplw finite groups with solvable 2-local subgroups" J.Algebra. 146. 412-426 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Hayashi and Y.Tanaka: "Amalgams of solvable groups" J.Fac.Sci.Univ.of Tolyo Sec.IA 39. 309-338 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Narain Gupta and Ken-Ichi Tahara: "The seventh and eighth Lie dimension subgroups" J.Pure and Appl.Algebra. 88. 107-117 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Ken-Ichi Tahara and Jiam Xiao: "On the seventh dimension subgroups, Japan." J.Math.New Ser.20. (to appear) (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Ken-Ichi Tahara and Jiam Xiao: "Supplements on our paper "On the seventh dimension subgroups"" Bull.Aichi Univ.of Educ.43. (to appear) (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] K.Gomi and M.Hayashi: "A pushing-up approach to the quasithin simple finite groups" J.Algebra. 146. 412-426 (1992)

    • Related Report
      1993 Annual Research Report
  • [Publications] M.Hayashi and Y.Tahaka: "Amalgams of solvable groups" J.Fac.Sci.Unio.of Tokyo Sec.A.I. 39. 309-338 (1992)

    • Related Report
      1993 Annual Research Report
  • [Publications] Norain Gupra and K.Tahara: "The seventh and eight Lie dimension subgroups" J Pur and Appl Algebra. 88. 107-117 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] K.Tahara and J.Xiao: "On the seventh Lie dimension subgroups" Japan.J.Math New.Sev.20. (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] K Tahara and J Xiam: "Supplements on our paper“On the seventh Lie dimension subgroups"" Bull.Aichi Unio.,Educ.43. 81-94 (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] Ken-Ichi Tahara and Jian Xiao: "On the seventh hie dimension subgroups" Japan.J.Math.

    • Related Report
      1992 Annual Research Report
  • [Publications] Ken-Ichi Tahara and Jian Xiao: "Suplemerts on our paper “On the seventh hie dcminsion subgroups"" Bull.Aichi Uniu of Edu.

    • Related Report
      1992 Annual Research Report
  • [Publications] Narain Gupta and Ken-Ichi Tahara: "The seventh and eighth hie dimension subgroups" J.Pare Appl Algebra.

    • Related Report
      1992 Annual Research Report
  • [Publications] Ken-Ichi Tahara and Jiar Xiao: "On the seventh hie dimension subgroups" to oppear.

    • Related Report
      1992 Annual Research Report
  • [Publications] KenーIchi Tahara: "On the rank of the hie powers of the of the auymentation ileal" Math.Proc.Cambridge Phil.Soc.

    • Related Report
      1991 Annual Research Report
  • [Publications] KenーIchi Tahara and Jian Xiao: "On the seventh Lie dimension subgrougs" J.Algebra.

    • Related Report
      1991 Annual Research Report

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Published: 1991-04-01   Modified: 2016-04-21  

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