1993 Fiscal Year Final Research Report Summary
Classification of ideals in integral group rings
| Project/Area Number |
03640041
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Aichi University of Education |
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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)
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| 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)
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| Project Period (FY) |
1991 – 1993
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| Keywords | Augmentation ideal / Lie power / Lie dimension subgroup |
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| 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.
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