Generalized Burnside rings and generalized prime graphs of finite groups
Project/Area Number 
17540028

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

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

CoInvestigator(Kenkyūbuntansha) 
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)

Keywords  finite group / prime graph / subgroup lattice / Burnside ring / simple group 
Research Abstract 
The concept of generalized prime graphs of finite group was introduced by AbeIiyori 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 Glattice 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
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l the triple (R(G),B(G),f) the oset 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 GPGconnected 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 2Frobenius 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 GruenbergKegel 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^sl) 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
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Research Products
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