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A study of representations and finite group actions realizing a given fixed point set

Research Project

Project/Area Number 14540084
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKYUSHU UNIVERSITY (2004)
九州芸術工科大学 (2002-2003)

Principal Investigator

SUMI Toshio  Kyushu University, Faculty of Design, Associate Professor, 大学院・芸術工学研究院, 助教授 (50258513)

Co-Investigator(Kenkyū-buntansha) IWASE Norio  Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (60213287)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsgap modules / gap group / finite group actions
Research Abstract

Let G be a finite group not of prime power order. For a prime p, we denote by O^p(G), called the Dress group of type p, the smallest normal subgroup of G with p-power index. The group G is said to be a gap group if there exists a G-module V such that dim V^<O^2(G)>=0 for any prime p and dim V^P>2dim V^H for any pair (P,H) of subgroups of G with some condition. Note that a Dress subgroup of a gap group G is not of prime power order.
I show that G is a gap group if and only if all subgroups L of G, possessing cyclic quotients L/O^2(G), are gap groups. Now assume that G/O^2(G) is cyclic. As viewing the series of normal groups
G=G_0〓G_1〓G_2〓【triple bond】〓G_k=O^2(G),[G_j,G_<j-1>]=2,
I obtained that G is a gap group if and only if each G_j(0<j<k) is a gap group. I define a subset E_j of 2-elements h of G_j\G_<j-1> by using a form of the centralizer C_G(h). We can easily decide whether the set E_j is empty or not, for example, letting j>1,E_j is not empty if there exists an element of G_j\G_<j-1> not of prime power order. It is a little bit complication to decide whether E_1 is empty. Then I showed that all E_j are nonempty if and only if G is a gap group. Furthermore, I obtained that if G×【triple bond】×G is a gap group,then so is G×G.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (10 results)

All 2004 2003 2002 Other

All Journal Article (9 results) Publications (1 results)

  • [Journal Article] 2-elements outside of the Dress subgroup of type 22004

    • Author(s)
      Toshio Sumi
    • Journal Title

      Transformation Group Theory and Surgery, RIMS Kokyuroku 1393

      Pages: 33-43

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Gap modules for semidirect product groups2004

    • Author(s)
      Toshio Sumi
    • Journal Title

      Kyushu Journal of Mathematics 58

      Pages: 33-58

    • NAID

      130000063078

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Implications of the Ganea condition2004

    • Author(s)
      Norio Iwase, Donald Stanley, Jeffrey Strom
    • Journal Title

      Algebraic and Geometric Topology 4

      Pages: 829-839

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Donald Stanley and Jeffrey Strom, Implications of the Ganea condition2004

    • Author(s)
      Norio Iwase
    • Journal Title

      Algebraic and Geometric Topology 4

      Pages: 829-839

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Implications of the Ganea condition2004

    • Author(s)
      Norio Iwase, Donald Stanley, Jeffrey Strom
    • Journal Title

      Algebr. Geom. Topol. 4

      Pages: 829-839

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Ganea conjecture and recent developments on the Lusternik-Schnirelmann category2004

    • Author(s)
      Norio Iwase
    • Journal Title

      Sugaku 56

      Pages: 231-296

    • Related Report
      2004 Annual Research Report
  • [Journal Article] L-S categories of simply-connected compact simple Lie groups of low rank2004

    • Author(s)
      Norio Iwase, Mamoru Mimura
    • Journal Title

      Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), Progr. Math. 215

      Pages: 199-212

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On finite groups possessing gap modules2003

    • Author(s)
      Toshio Sumi
    • Journal Title

      Transformation Group Theory and Related Topics, RIMS Kokyuroku 1343

      Pages: 99-104

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Nonsolvable general linear groups are gap groups2002

    • Author(s)
      Toshio Sumi
    • Journal Title

      Transformation groups from new points of view, RIMS Kokyuroku 1290

      Pages: 31-41

    • NAID

      110000166783

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Toshio SUMI: "GAP MODULES FOR SEMIDIRECT PRODUCT CROUPS"Kyushu Journal of Mathematics. 58. 1-26 (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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