On Fusion Systems of Component Type
On Fusion Systems of Component Type
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论构件型融合系统
DOI：
10.1090/memo/1236
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
M. Aschbacher
中科院分区：
文献类型：
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作者：
M. Aschbacher
介绍。本系列讲座涉及局域群论与融合系统理论之间的相互作用，重点关注使用融合系统来简化有限单群分类定理证明的部分可能性。就我们的目的而言，有限单群的分类始于 GorensteinWalter 二分定理（参见 [ALSS]），该定理表示每个 2 阶至少 3 的有限群 G 要么是分量类型，要么是特征 2 类型。从 2-局部结构的角度来看，这将有限群划分为奇数和偶数特征的群。我们几乎只关心具有奇怪特征的组：组件类型组。然而，乌尔里希·迈尔弗兰肯菲尔德的讲座可以被认为是关注具有均匀特征的群体。在饱和融合系统 F 的情况下，相对于 GorensteinWalter 二分法的情况更好：F 要么是特征 p 型，要么是组件类型，与等级无关。此外，饱和融合系统的二分定理比群定理更容易证明；事实上，一旦饱和融合系统 F 的广义拟合子系统 F*(F) 的概念到位，并且建立了 F*(F) 的合适属性（包括 E 平衡），融合系统的二分定理的证明就很容易了。但更重要的是，使用组件类型的 2-fusion 系统似乎比使用组件类型的组更容易。这是因为在分量类型的群G中，G的2局部子群H可能具有非平凡核，其中H的核是H的奇数阶的最大正规子群O(H)。这些核心的存在给组件类型组的分析带来了大问题。如果能够证明 B 猜想，这些问题就可以最小化，该猜想表示，在一个简单群中，
Introduction. This series of lectures involves the interplay between local group theory and the theory of fusion systems, with the focus of interest the possibility of using fusion systems to simplify part of the proof of the theorem classifying the finite simple groups. For our purposes, the classification of the finite simple groups begins with the GorensteinWalter Dichotomy Theorem (cf. [ALSS]) which says that each finite group G of 2-rank at least 3 is either of component type or of characteristic 2-type. This supplies a partition of the finite groups into groups of odd and even characteristic, from the point of view of their 2-local structure. We will be concerned almost exclusively with the groups of odd characteristic: the groups of component type. However Ulrich Meierfrankenfeld’s lectures can be thought of as being concerned with the groups of even characteristic. In the case of a saturated fusion system F , the situation vis-a-vis the GorensteinWalter dichotomy is nicer: F is either of characteristic p-type or component type, irrespective of rank. Further the Dichotomy Theorem for saturated fusion systems is much easier to prove than the theorem for groups; indeed once the notion of the generalized Fitting subsystem F ∗(F) of a saturated fusion system F is put in place, and suitable properties of F ∗(F) are established, including E-balance, the proof of the Dichotomy Theorem for fusion systems is easy. But of more importance, it seems easier to work with 2-fusion systems of component type than with groups of component type. This is because in a group G of component type, a 2-local subgroup H of G may have a nontrivial core, where the core of H is the largest normal subgroup O(H) of H of odd order. The existence of these cores introduces big problems into the analysis of groups of component type. These problems can be minimized if one can prove the B-Conjecture, which says that, in a simple group,