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Generalized Burnside rings and generalized prime graphs of finite groups

Research Project

Project/Area Number 17540028
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

IIYORI Nobuo  Yamaguchi University, Education, Assistant Professor, 教育学部, 助教授 (00241779)

Co-Investigator(Kenkyū-buntansha) KITAMOTO Takuya  Yamaguchi University, Education, Assistant Professor, 教育学部, 助教授 (30241780)
KUTAMI Mamoru  Yamaguchi University, science, Professor, 理学部, 教授 (80034734)
MIYAZAWA Y.  Yamaguchi University, science, Assistant Professor, 理学部, 助教授 (60263761)
MATUNO Yosimasa  Yamaguchi University, engineering, Professor, 工学部, 教授 (30190490)
NISHIYAMA T.  Yamaguchi University, engineering, Assistant Professor, 工学部, 助教授 (60333241)
Project Period (FY) 2005 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2006: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2005: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsfinite group / prime graph / subgroup lattice / Burnside ring / simple group
Research Abstract

The concept of generalized prime graphs of finite group was introduced by Abe-Iiyori at 2000 and some elementary properties of the graphs were studied. Iiyori showed some relations between the connected components of the generalized prime graph of a finite group G and a decomposition of the generalized Burnside ring of G by its ideals at 2004. The purposes of this research are to make the relations between generalized prime graphs and generalized Burnside rings clear, and to study finite groups as its applications.
Let G be a finite group and let denote Sub(G) be the totality of solvable subgroups of G. Then Sub(G) is a G-lattice with respect to the containment relation. Let R(G) be the formal sum of Sub(G) with rational integer coefficients. This R(G) forms a ring under the multiplication AB=(the meet of A and B). The generalized Burnside ring B(G) can be defined as a sub ring of R(G). Let f be the mapping of R(G) to B(G) which is defined by f(x)=the sum of x^g(gin G) for each x.We cal … More l the triple (R(G),B(G),f) the o-set triple. Our first main result is the following: let G be a finite group. Let X be a finite simple group which is not isomorphic to neither C_n (q) nor B_n (q) (q'odd). ffthe oset triples of'G and X are isomorphic each other, then G and Xare isomorphic.
Two ideals I, J of B(G) are said to be disconnected if they satisfies (D1) I+J=B(G) and (D2) the meet of I and J is generated by the subgroup 1. By the same manner we define the disconnectedness among k ideals I_1,..,I_k. The second main result is the following : Let $G$ be a finite group. The number of GPG-connected components of B(G) equals (the number of connected components of prime graph of G)-(the number of connected components of the prime graph of G of 2-Frobenius type with respect to solvability).
We also studied some properties of prime graphs of finite group to observe structures of generalized prime graphs and generalized Burnside rings. Our Third main results is the next: Let G be a finite group of even order. If an odd prime divisor p of the order of G is not joined to 2 by edge directly, then G has a chain of normal subgroups of G like the Gruenberg-Kegel chain. Moreover if a Sylow psubgroup ofG is not a cyclic group, then G has at most one non abelian simple factor and the factor is isomorphic to PSL_2 (p^a)〜(a>1), 〜PSL_3 (2^s)〜(3l l2^s-l) or PSU_3 (2^t)〜(3l l 2^t+1).
We also showed interesting properties of generalized prime graphs and generalized Burnside rings, which can be found in "A Generalization of Prime Graphs of Finite Groups II"(submitting) and in other papers of ours(submitting). Less

Report

(3 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • Research Products

    (2 results)

All 2006

All Journal Article (2 results)

  • [Journal Article] 素数グラフ、一般素数グラフとその周辺2006

    • Author(s)
      飯寄 信保
    • Journal Title

      第51回代数学シンポジウム報告集 51

      Pages: 65-78

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Prime graphs, generalized prime graphs and their related topics2006

    • Author(s)
      Nobuo Iiyori
    • Journal Title

      Proc.of 51th Symposium of Algebra (Tokyo University)

      Pages: 65-78

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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