Generalized Burnside rings and generalized prime graphs of finite groups

Abe-Iiyori在2000年提出了有限群的广义素图的概念，并研究了它的一些基本性质。2004年，Iiyori证明了有限群G的广义素图的连通分支与G的广义Burnside环的理想分解之间的一些关系。本文的目的是明确广义素图与广义Burnside环之间的关系，并研究有限群的应用.设G是有限群，记Sub(G)是G的可解子群的全体，则Sub(G)是关于包含关系的G-格.设R(G)是Sub(G)具有有理整数系数的形式和。这个R(G)在乘法AB=(A和B的交集)下形成一个环。广义Burnside环B(G)可以定义为R(G)的子环。设f是R(G)到B(G)的映射，其定义为f(X)=每个x的x^g(Gin G)之和，我们称其为…更多的是L的三元组(R(G)，B(G)，f)的o-set三元组。我们的第一个主要结果如下：设G是有限群。设X是一个有限单群，它既不同构于C_n(Q)，也不同构于B_n(Q)(q‘奇异)。如果G和X的Oset三元组彼此同构，则G和X同构。如果B(G)的两个理想I，J满足(D_1)I+J=B(G)和(D_2)I和J的交是由子群1生成的，则称它们是不连通的。用同样的方法定义了k个理想I_1，…，I_k之间的不连通性。第二个主要结果如下：设$G$是有限群。B(G)的GPG-连通分支个数等于(G的素图的连通分支个数)-(2-Frobenius型素图的连通分支个数)。我们还研究了有限群的素图的一些性质，以观察广义素图和广义Burnside环的结构。我们的第三个主要结果是：设G是偶数阶有限群。如果G的阶奇素数p不是通过边直接与2相连，则G有一个像Gruenberg-Kegel链一样的正规子群链。此外，如果G的一个Sylow子群不是循环群，则G至多有一个非交换单因子，且该因子同构于PsL_2(p^a)~(a&gt；1)，~PsL_3(2^S)~(3l L2^S-L)或PsU_3(2^t)~(3l L 2^t+1)。我们还给出了广义素图和广义Burnside环的有趣性质，这些性质可以在提交的《有限群素图的推广II》和我们的其他论文中找到。较少

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