Subgroup lattices and prime graphs of finite groups

• Project/Area Number: 16K05062
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Yamaguchi University
• Principal Investigator: Iiyori Nobuo 山口大学, 教育学部, 教授 (00241779)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 有限群 / 部分群束 / クイバ / 素数グラフ / ブラウワー指標 / 表現論 / クイバー / ホモロジー / 群指標 / quiver / 単純群

Outline of Final Research Achievements

The main results obtained in this study are as follows.
(1) We have found an algorithm to compute homology groups of subgroup lattices of general linear groups.(2) Letπbe the set of prime divisors of the order of a finite group G and let π＝π1Uπ2 be a disjoint union. Then we found the relation among the homology groups of nilpotent π-subgroup lattice and the homology groups of nilpotent πi-subgroups lattices(i=1,2) via Mayer-Vietoris sequence.(3) Let p and q be prime numbers. We defined a generalized Cartan matrix which describes relations between Brauer p-characters and Brauer q-characters (an ordinary Cartan matrix describes relations between ordinary characters and Brauer p-characters.) and we show it is a regular matrix.

Academic Significance and Societal Importance of the Research Achievements

通常、有限群の表現論は主に単一の標数に関する表現、またはその指標について考察されるか、或は、２つの標数0、p（素数）の場合について考察がなされてきた。本研究では複数の素数に対する指標間の関係を部分群束が統率すると考え、クイバー（部分群束）の表現の視点から異なる標数のモジュラー指標達の関係を調べることで、複数の標数のモジュラー指標間においてもカルタン行列、フロベニウスの相互律等の重要な性質・概念の類似を見出すことができた。このことから本研究は群の研究における新しい視点を与えるものと考えられ、今後の有限群の研究に寄与することが期待できる。

Research Products

• [Journal Article] Quiver representations, group characters, and prime graphs of finite groups (2019)
  • 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
  • Peer Reviewed / Open Access
• [Journal Article] Homology of the complex of all non-trivial nilpotent subgroups of a finite non-solvable group (2019)
  • 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
  • Peer Reviewed / Open Access
• [Journal Article] Quiver representations, group characters and prime graphs of finite groups. (2019)
  • Author(s): Nobuo Iiyori and Masato Sawabe
  • Journal Title: Tokyou Journal of Mathematics Volume: 42 Vol 2
  • Peer Reviewed
• [Journal Article] Partially ordered sets of non-trivial nilpotent π-subgroups II (2017)
  • 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
  • Peer Reviewed / Open Access
• [Presentation] Homology of the complex of all non-trivial nilpotent subgroups of a finite non-solvable group (2019)
  • Author(s): 飯寄信保, 澤辺正人
  • Organizer: 有限群のコホモロジー論とその周辺
• [Presentation] クイバーの表現と有限群の部分群束 (2017)
  • Author(s): 飯寄信保
  • Organizer: 日本数学会