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Amalgam methods applied to the structure of the finite simple groups of characteristic 2 type

Research Project

Project/Area Number 10640033
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionOita University

Principal Investigator

TANAKA Yasuhiko  Oita University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (70244150)

Co-Investigator(Kenkyū-buntansha) SATO Shizuka  Oita University, Faculty of Engineering, Professor, 工学部, 教授 (20040719)
OHKOHCHI Shigemi  Oita University, Faculty of Engineering, Professor (70128338)
Project Period (FY) 1998 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2000: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1999: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywordsamalgams / solvable 2 local subgroups / characteristic 2 type / finite simple groups / revision / classification / prime graphs / rivision / solvable 2 local subgroups
Research Abstract

The purpose of this project is to classify the finite simple groups satisfying a certain condition (A). The actual work is stated in the following three-step procedure. 1. Classify the isomorphism classes of the amalgams (X, S, Y) of a finite 2-group S and finite groups X, Y containing S as their common Sylow 2-subgroup satisfying the conditions : (1) both O_2(X) and O_2(Y) are nonidentity ; (2) O_2(X, Y) is identity : (3) an additional condition may be fulfilled. 2. For a finite simple group G satisfying the condition (A), find an amalgam (X, S, Y) of subgroups S, X, Y of G satisfying the above three conditions. 3. Classify the finite simple groups G satisfying the condition (A) by judging which isomorphism class in the first step the amalgam of G found in the second step is belonging to.
Basically, our result is divided into three categories. First of all, when the condition (A) is said to be 'simple groups of characteristic 2 type all of whose 2-local subgroups are solvable,' we study the third step above, and get a new proof of the classification of such simple groups. Our proof is much shorter and conceptually easier than the existing one. Next, more generally, when the condition (A) is 'simple groups of characteristic 2 type,' we study the third step above, and determine the structure of quadratic module of finite simple groups. In the analysis of 2-local subgroups of finite simple groups of characteristic 2 type, we often encounter modules over GF(2) having quadratic 2-subgroups. So we can expect that this result will apply to the study of the third step above. Finally, when the condition (A) is 'simple groups of characteristic 2 type,' we also study the geometric structure of the prime graphs of finite simple groups. Though the importance of the connected components of only odd primes was already well known, we focus attention on its connected components containing the prime 2 and even local subgroups instead of 2-local subgroups.

Report

(5 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • 1998 Annual Research Report

Research Products

(8 results)

All Other

All Publications (8 results)

  • [Publications] Yasuhiko Tanaka: "Even local subgroups of a finite simple group"Kokyuroku, RIMS, Kyoto University. 1214. 4-10 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Makoto Hayashi and Yasuhiko Tanaka: "On the finite simple groups all of whose 2-local subgroups are solvable"Journal of Algebra. 210. 365-384 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yasuhiko Tanaka: "Even local subgroups of a finite simple group"Kokyuroku RIMS, Kyoto University. 1214. 4-10 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shizuka Sato: "On the ring of constants of a derivation of R[x, y]"Reports, Fac. Eng. Oita University. 39. 13-16 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yasuhiko Tanaka: "Even local subgroups of a finite simple group"京都大学数理解析研究所講究録. 1214. 4-10 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 田中康彦: "単純群の分類問題のその後"京都大学数理解析研究所. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] Yasuhiko Tanaka: "On quadratic 2-groups"Preprint(http://www.csis.oita-u.ac.jp/〜tanaka/maths/).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Hayashi and Y.Tanaka: "On the finite simple groups all of whose 2-local subgroups are solvable" Journal of Algebra. (印刷中).

    • Related Report
      1998 Annual Research Report

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Published: 1998-03-31   Modified: 2016-04-21  

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