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Nilpotent subgroup complexes of finite groups and associated quiver representations

Research Project

Project/Area Number 17K05161
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionChiba University

Principal Investigator

Sawabe Masato  千葉大学, 教育学部, 教授 (60346624)

Project Period (FY) 2017-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords有限群 / 単体複体 / 表現論 / ポセット / 半順序集合 / 指数 / 表現 / 部分群複体 / ホモロジー / 類関数 / 群複体
Outline of Final Research Achievements

A family of subgroups of a finite group can be regarded as a partially ordered set, an order complex, or a quiver with respect to the inclusion relation among the subgroups. First, we determined the homology group of the complex of non-trivial nilpotent subgroups of a finite non-solvable group. Second, we proposed the quiver representation associated to the complex of the totality of subgroups, and developed the basis for it. Third, we introduced the concept of d-cover for a certain complex of nilpotent subgroups, and in particular we characterized 1-cover. In addition, we proved the group-theoretic nature derived from the 2-cover.

Academic Significance and Societal Importance of the Research Achievements

世の中のあらゆるところに存在する対称性を抽象化した数学的対象は群と呼ばれる。群はある種の代数系であるが、群が有する沢山の部分群におけるその配置を幾何学的に考察し、その情報を元の群の性質にフィードバックさせることが本研究の特徴である。学術的に本研究成果は有限単純群の統一的理解への示唆を与えるものである。また社会的は一般的な対称性の数学的理解を与えるものである。

Report

(6 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (10 results)

All 2022 2020 2019 2017

All Journal Article (5 results) (of which Peer Reviewed: 5 results,  Open Access: 5 results) Presentation (5 results) (of which Invited: 3 results)

  • [Journal Article] Quiver representations, group characters, and prime graphs of finite groups2020

    • Author(s)
      Nobuo Iiyori and Masato Sawabe
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 43

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Homology of the complex of all non-trivial nilpotent subgroups of a finite non-solvable group2019

    • Author(s)
      Nobuo Iiyori and Masato Sawabe
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 42 Issue: 1 Pages: 113-120

    • DOI

      10.3836/tjm/1502179264

    • Related Report
      2019 Research-status Report 2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Quiver representations, group characters, and prime graphs of finite groups2019

    • Author(s)
      Nobuo Iiyori and Masato Sawabe
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 42 Issue: 2 Pages: 497-523

    • DOI

      10.3836/tjm/1502179297

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Partially ordered sets of non-trivial nilpotent π-subgroups II2017

    • Author(s)
      Nobuo Iiyori and Masato Sawabe
    • Journal Title

      Topology and its Applications

      Volume: 231 Pages: 197-218

    • DOI

      10.1016/j.topol.2017.09.011

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Homology of a certain associative algebra2017

    • Author(s)
      Nobuo Iiyori and Masato Sawabe
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 46 Issue: 2 Pages: 227-256

    • DOI

      10.14492/hokmj/1498788019

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] d-covers of posets of nilpotent subgroups2022

    • Author(s)
      澤邉正人
    • Organizer
      有限群のコホモロジー論とその周辺
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] 部分群複体とその周辺2022

    • Author(s)
      澤邉正人
    • Organizer
      愛媛代数セミナー
    • Related Report
      2021 Annual Research Report
  • [Presentation] Homology of the complex of all non-trivial nilpotent subgroups of a finite non-solvable group2019

    • Author(s)
      飯寄信保 澤邉正人
    • Organizer
      有限群のコホモロジー論とその周辺
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Nilpotent π-subgroups and gluing complexes2017

    • Author(s)
      飯寄信保 澤邉正人
    • Organizer
      代数的組合せ論および有限群・頂点作用素代数とその表現の研究
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] ベキ零部分群複体のホモロジーについて2017

    • Author(s)
      澤邉正人
    • Organizer
      群論特別セミナー
    • Related Report
      2017 Research-status Report

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Published: 2017-04-28   Modified: 2023-01-30  

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