1990 Fiscal Year Final Research Report Summary
| Project/Area Number |
63540034
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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ーIchi Department of Mathematics, Professor, 教育学部, 教授 (00024026)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Osamu Aichi Univ. Educ., Professor, 教育学部, 教授 (30024011)
IKEDA Yoshiaki Aichi Univ. Educ., Professor, 教育学部, 教授 (00022640)
FURUKAWA Yasukuni Aichi Univ. Educ., Professor, 教育学部, 教授 (90024033)
HAYASHI Makoto Aichi Univ. Educ., Assis. Professor, 教育学部, 助教授 (40109369)
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| Project Period (FY) |
1988 – 1990
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| Keywords | Integral group ring / Augmentation ideal / Lie dimension subgroup / Lie dimension subgroup problem / Lower central series |
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| Research Abstract |
Let G be a group with lower central series G = G_1 * G_2 * ・・・ * G_n * G_<n+1> * ・・・. Denote by ZETAG the integral group ring of G over ZETA the ring of rational integers, and by DELTA (G) the augmentation ideal of ZETAG. For any elements alpha, betaepsilonZETAG, we denote (alpha, beta) =alphabeta-betaalpha. We define inductively Lie powers of DELTA(G) as follows ; DELTA^<(1)>(G)=DELTA(G), DELTA^<(n)>(G)=(DELTA^<(nー1)>(G), DELTA(G))ZG = <(alpha, beta)|alphaepsilonDELTA^<(nー1)>(G), betaepsilonDELTA(G)>ZETAG. We define the n-th Liedimension subgroup D_<(n)>(G)= G*(1+DELTA^<(n)>(G)). Then Lie Dimension Subgroup Problem means a characterization of D_<(n)>(G). one of the important results we learned up to now is the following :
Theorem 1 (R. Sandlin). For any n with 1*n*6, it follows D_<(n)>(G)=G_n. At first, we get the following result to get an extension of Theorem 1.
Theorem 2. Let G be a group such that G_2/D_3 has finite exponent. Then rank_<ZETA> DELTA^<(n)>(G) = rank_<ZETA> DELTA^<(2)>(G), for any n*2. In particular, G is a finite group, then rank_<ZETA> DELTA^<(1)>(G) = |G| - 1 rank_<ZETA> DELTA^<(n)>(G)2 DELTA^<(n)>(G) = |G| - |G/G_2| = |G/G_2|(|G_2| - 1 ). Next, we have the following to get information on D_<(n+1)>(G) when we have D_<(n)>(G) = G_n.
Theorem 3. For any n*1, there is a homomorphism PSI_n : G_n/G_<n+1> -> DELTA^<(n)>(G)/DELTA^<(n+1)>(G) such that G_n*(1 + DELTA^<(n+1)>(G)) = D_<(n+1)> (G) <=> PSI_n : injective.
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