Nilpotent subgroup complexes of finite groups and associated quiver representations

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K05161
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Chiba 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
• Keywords: 有限群 / 単体複体 / 表現論

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.

Free Research Field

有限群

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。