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

汞齐法在特征2型有限单群结构中的应用

基本信息

批准号：
10640033
负责人：
TANAKA Yasuhiko
金额：
$ 1.92万
依托单位：
Oita University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640033/
关键词：
amalgams solvable 2 local subgroups characteristic 2 type finite simple groups revision classification prime graphs rivision solvable 2 local subgroups

项目摘要

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.

本课题的目的是对满足一定条件（A）的有限单群进行分类。实际工作在以下三个步骤中说明。1.对有限2-群S与包含S作为公共Sylow 2-子群的有限群X，Y的混合（X，S，Y）的同构类进行分类，它们满足以下条件：（1）O_2（X）和O_2（Y）都是非恒等的;（2）O_2（X，Y）是恒等的;（3）可满足一个附加条件。2.对于满足条件（A）的有限单群G，求满足上述三个条件的子群S，X，Y的混合（X，S，Y）。3.对满足条件（A）的有限单群G，通过判断第二步中得到的G的混合属于第一步中的同构类来进行分类，基本上我们的结果分为三类.首先，当条件（A）为“特征2型单群的所有2-局部子群可解”时，我们研究了上述第三步，得到了这类单群分类的一个新的证明。我们的证明是更短，概念上比现有的更容易。其次，更一般地，当条件（A）是“特征2型单群”时，我们研究上面的第三步，并确定有限单群的二次模的结构。在特征2型有限单群的2-局部子群的分析中，经常会遇到GF（2）上的模具有二次2-子群。因此，我们可以预期，这一结果将适用于上述第三步的研究。最后，当条件（A）为“特征2型单群”时，我们还研究了有限单群素图的几何结构。虽然奇素数连通分支的重要性已经众所周知，我们把注意力集中在它的连通分支包含素2和偶局部子群，而不是2-局部子群。