The Solomon system FSol(3) does not occur as fusion system of a 2-block
The Solomon system FSol(3) does not occur as fusion system of a 2-block
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所罗门系统 FSol(3) 不会作为 2 块的融合系统出现
DOI：
10.1016/j.jalgebra.2005.09.026
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发表时间：
2006
影响因子：
0.9
通讯作者：
R. Kessar
中科院分区：
文献类型：
--
作者：
R. Kessar
设 p 为素数。 Puig 在局部分块理论的背景下引入了有限 p 群上的融合系统（在 [29] 中称为完整 Frobenius 系统，在 [6] 中称为饱和融合系统）。对于每对 (G, P)，其中 G 是有限群，P 是 G 的 Sylow p 子群，与 P 上的融合系统 FG (P) 相关联，称为 G 的 p 融合系统，它反映 G 的 p 融合模式。类似地，对于每个四元组 (H, b, Q, e)，其中 H 是有限群，b 是 H 的 p 块，(Q, e) 是最大 b-布劳尔对（即 Q 是 G 的缺陷群，并且e是对应于b)的CG(Q)的p块，与P上的融合系统F(H，b)(Q，e)相关联，称为b的融合系统。一个有趣的融合系统是所罗门系统 FSol (3)，它是经典群 Spin7 (3) 的 Sylow 2 子群上的系统。 FSol (3) 在[22]中被定义和研究，但所罗门在[32]中隐含地考虑了。 Solomon 的结果意味着 FSol (3) 是“奇异的”，即不存在具有 FSol (3) 作为 2 融合系统的有限群。结果，所罗门证明了唯一具有与康威群同构的 Sylow 2 子群的有限单群。 3 是。 3、有限单群分类的重要一步。在[4]中，Benson构造了一个拓扑空间，该空间是具有FSol（3）作为2-融合系统的有限群的2-完备分类空间（如果存在这样的群），他预测这种构造是一般理论的一个实例，这一见解在[6]中的p-局部群的Broto-Levi-Oliver理论中得到了实现。
Let p be a prime number. Fusion systems (referred to as full Frobenius systems in [29], and as saturated fusion systems in [6]) on finite p-groups, were introduced by Puig in the context of local block theory. To each pair (G, P) where G is a finite group and P is a Sylow p-subgroup of G, is associated a fusion system FG (P) on P called a p-fusion system of G, which reflects the p-fusion pattern of G. Similarly, to each quadruple (H, b, Q, e) where H is a finite group, b is a p-block of H, and (Q, e) is a maximal b-Brauer pair (that is Q is a defect group of G and e is a p-block of CG (Q) in correspondence with b) is associated a fusion system F (H, b)(Q, e) on P, called a fusion system of b. An interesting fusion system is the Solomon system FSol (3) which is a system on a Sylow 2-subgroup of the classical group Spin7 (3). FSol (3) is defined and studied in [22], but was implicitly considered by Solomon in [32]. Solomon’s results imply that FSol (3) is “exotic,” that is there is no finite group having FSol (3) as a 2-fusion system. As a consequence, Solomon proved that the only finite simple group having a Sylow 2-subgroup isomorphic to that of the Conway group. 3 is. 3, an important step in the classification of finite simple groups. In [4], Benson constructed a topological space which would have been the 2-completed classifying space of a finite group having FSol (3) as 2-fusion system (had such a group existed), and he predicted that this construction was an instance of a general theory, which insight was realized in the Broto–Levi–Oliver theory of p-local groups in [6].