Generalized Burnside rings and generalized prime graphs of finite groups

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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