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1990 Fiscal Year Final Research Report Summary

Lle Dimension Subgroups

Research Project

Project/Area Number 63540034
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ーIchi  Department of Mathematics, Professor, 教育学部, 教授 (00024026)

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)
Project Period (FY) 1988 – 1990
KeywordsIntegral group ring / Augmentation ideal / Lie dimension subgroup / Lie dimension subgroup problem / Lower central series
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.

  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] I.B.S.Passi,Sucheta and KenーIchi Tahara: "Dimension subgrops and Schur multiplicatorーIII" Japan.J.Math.New Ser.13. 371-379 (1987)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] KenーIchi Tahara: "Problems on integral group rings" Essays of Prof.Jang II Um on his 60 birth day Department of Mathematics,Pusan National University. 293-299 (1988)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Makoto Hayashi: "A note on amalgams" Hokkaido Math.J.19. 431-434 (1990)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] KenーIchi Tahara: "On the rank of the Lie powers of the augmentation ideal"

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I. B. S. Passi, SUcheta and Ken-ichi Tahara: "Dimension subgroups and Schur multiplicator-III" Japan. J. Math. New Ser.13. 371-379 (1987)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kenーichi, Tahara: "Problems on integral group rings" Essays of Prof. Jang Il Um on his 60th birthday Department of Mathematics, Pusan National University.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Makoto, Hayashi: "A note on amalgams" Hokkaido Math. J.19. 431-434 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kenーichi, Tahara: "On the rank of the Lie powers of the augmentation ideal"

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1993-08-12  

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