Finite simple groups of parabolic characteristic p

修正有限单群的分类实际上有两种不同的方法。一个归功于G.Gorenstein，R.Lyons和R.所罗门(GLS)，他们将给出遵循经典直线的证明。最近，M.Aschbacher在核聚变系统的帮助下开始了一种新的方法。到目前为止，后一种方法只适用于组件类型的组。与之平行的还有U·梅尔法恩肯菲尔德的工作，B.Stellmacher和G.Stroth(MSS)提出了一种新的局部特征群的分类方法。这项工作的主要目的是在MSS的结果和其它两个方案之间建立一座桥梁，以便它们可以在那里使用。在最初的分类中，对成分类型群和局部特征群2进行了明确的细分。GLS现在谈论偶型群而不是局部特征群2。Aschbacher在他的程序中将问题简化为考虑包含极大阿贝尔子群的局部子群是特征2的群。一个关键可以是结果归因于K.Magaard和G.Stroth对非抛物线特征2的偶型群的分类。为了使MSS的结果对GLS可用，证明抛物线特征2的这些结果就足够了，也就是说，包含Sylow 2-子群的局部子群具有局部特征2。作为副作用，这也将为使用Aschbacher程序中的结果提供桥梁。更确切地说，我们将在MSS中证明抛物线特征2的群的结构定理。此外，识别出现的单群也应该仅假设抛物线特征2。特别地，我们必须处理包含奇指数子群$H$的群$G$，它是特征2中Lie型群的自同构群。只要假定抛物线特征2，我们将确定所有这样的对(G，H)。

There are actually two different approaches to revise the classification of the finite simple groups. One due to G. Gorenstein, R. Lyons and R. Solomon (GLS) which will give a proof following the classical line. Recently M. Aschbacher started a new approach with the help of fusion systems. So far the latter approach just work for the groups of component type. Parallel there is the work of U. Meierfarnkenfeld, B. Stellmacher and G. Stroth (MSS) which aim to gve a new classification of the groups of local characteristic p. The main aim of this project is bulid a bridge between the results of MSS and the other two projects such that they can be used there.In the original lassification there was a clear subdivision in groups of component type and groups of local characteristic 2.GLS now speaks of groups of even type instead of groups of local characteristic 2. Aschbacher in his program reduces the problem to consider groups in which local subgroups containing a maximal abelian subgroup are of characteristic 2. A key could be a result due to K. Magaard and G. Stroth which classifies the groups which are of even type but not of parabolic characteristic 2. To make the results of MSS accessible for GLS it would suffice to prove these results for parabolic characteristic 2, i.e. local subgroups containing a Sylow 2-subgroup are of local characteristic 2. As a side effect this would also give the bridge to use the results in the Aschbacher program.The aim of this project is to achieve exactly this goal. More precise we will prove the structure theorem in MSS for groups of parabolic characteristic 2. Furthermore identifying the simple groups showing up should also be done by just assuming parabolic characteristic 2. In particular we have to deal with groups $G$ containing a subgroup $H$ of odd index, which is an automorphism group of a group of Lie type in characteristic two. Just assuming parabolic characteristic 2 we will determine all such pairs (G,H).