Project/Area Number  09640035 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Osaka Kyoiku University 
Principal Investigator 
YOSHIARA Satoshi Fac.Education, Osaka Kyoiku University Professor, 教育学部, 教授 (10230674)

CoInvestigator(Kenkyūbuntansha) 
HIRAGI Akira Fac.Education, Osaka Kyoiku University Lecturer, 教育学部, 講師 (90294181)
KITAMURA Kazuo Fac.Education, Osaka Kyoiku University Professor, 教育学部, 教授 (30030381)
ITO Tasuro Kanazawa University Fac. Science Professor, 理学部, 教授 (90015909)
BABA Yoshitomo Fac.Education, Osaka Kyoiku University Lecturer, 教育学部, 講師 (10201724)
NAKAGAWA Nobuo Kinki University Fac.Science & Tech.Assistant Professor, 理工学部, 助教授 (10088403)

Project Fiscal Year 
1997 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥3,700,000 (Direct Cost : ¥3,700,000)
Fiscal Year 2000 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1999 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1997 : ¥1,000,000 (Direct Cost : ¥1,000,000)

Keywords  radical subgroup / centric / EGQ / Yfamily / dimensional dual hyperoval / semibiplane / Dade conjecture / homotopy equivalence / Dada conjecture / O'Nan群 / Dade予想 / 高次元双対弧 / 高次元双対超卵型 / 一般化された四辺形 / 距離正則グラフ / pcentric / モンスター / フィッシャー群 / GQ / 拡大双対極空間 / 高次元dual arc 
Research Abstract 
The research of the head investigator on radical subgroups (Project(b)) clarifies the mathematical significance of the notion of centric radical subgroups, and shows the efficiency of some methods in algebraic topology to study group complexes. Furthermore, starting with the classification of group complexes of low rank (Part of Project(a)), he found new combinatorial objects, called Yfamilies and dimensional dual hyperovals, which can be thought of as higher dimensional analogues of quadrics in projective planes. His recent researches contribute to establish their importance as research subjects in finite geometry and combinatorics. Further details on Project (a) : The notion of Yfamilies and dimensional dual hyperovals was found as well as the following machinery : their affine expansions give EGQ (extended generalized quadrangle)s and semibiplanes, which are complexes extending buildings and affine spaces of low rank. Motivated by the research of the head investigator, the classification of those objects has been now actively investigated as well as the related combinatorial objects, such as distance regular graphs. On Project (b) : (l) It is now convincing that minimal complexes homotopically equivalent to those of chains of centric radical subgroups are main objects to investigate. Examinations of several examples by the head investigator contribute to the recent theory of Makoto Sawabe on homotopy equivalence which covers many classes of geometries. In future, many researches in this area will be devoted to generalizing his results. (2) All radical subgroups of every sporadic simple groups except the Monster and the baby Monster for p=2 are determined.(In the exceptional cases, the results are believed to complete.) This gives the radical chains in many sporadic groups, and it makes possible to delete many chains unnecessary to examine for verifying the Dade conjecture.
