A Simple Subgroup of M? and E8(3)
A Simple Subgroup of M? and E8(3)
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M 的简单子群？
DOI：
10.1112/blms/8.2.161
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发表时间：
1976
影响因子：
0.9
通讯作者：
P. E. Smith
中科院分区：
文献类型：
--
作者：
P. E. Smith
本注记的目的是宣布在1974年5月构造一个新的阶为90,745,943,887,872,000=215的单群E。310.53.72.13.19.31在剑桥由JG Thompson和本作者所著。这是JG·汤普森对假设的怪物群M的某些调查的结果吗？B·Fischer，推测M？3阶的合适元素是C3xE。让我在开始时说，我参与这项工作的是计算机构造；数学是汤普森的工作。E的对合a的正规化化子是21+8A9，即C=C GB(A)包含一个29阶的正规超特殊子群H，使得C/H GB A9\正是根据这一性质，Thompson构造了E。从E的附加的特征标表(主要是Thompson和JH Conway的工作)可以看出，一个忠实的普通表示的最低程度是248；让它提供字符X。可以使用各种论元来表明这可以被假定为一个整数表示；也就是说，存在一个由E稳定的248维格L*。模所有素数的表示的不可约性表明L*可以被尺度化为单模的。我们将构造一个248维空间的变换群E*，它将被计算机证明为E^E*。设F=O2(CE(AJ)nO2(C GB(JS)，D=IVE(F)。F是25阶初等交换，D/F SL 2(5)且D不在F上分裂。Dempwolff在文[2]中证明了，如果存在这样的扩张，则它是唯一的。相应地，我们称D(我们显式构造的)为Dempwolff群。复数域上D的每个忠实表示都有^248次，存在唯一的248次忠实表示(J)，E的248次表示必须限制到它。Thompson构造了复李代数的一个分解
The purpose of this note is to announce the construction in May 1974 of a new simple group E of order 90, 745, 943, 887, 872, 000= 215. 310. 53. 72. 13.19. 31 at Cambridge by JG Thompson and the present author. This is the result of certain investigations by JG Thompson into the hypothetical monster group M? of B. Fischer; it is conjectured that the centralizer in M? of a suitable element of order 3 is C3 x E. Let me say at the outset that my involvement in this work has been the computer construction; the mathematics is Thompson's work. The ceritralizer of an involution a of E is 21+ 8 A9, ie C= C£(a) contains a normal extra-special subgroup H of order 29 such that C/H£ A9\it is in terms of this property that Thompson constructs E. It will be seen from the appended character table of E (chiefly the work of Thompson and JH Conway) that the least degree of a faithful ordinary representation is 248; let this afford the character X. Various arguments may be employed to show that this may be assumed to be an integer representation; that is, there exists a 248-dimensional lattice L* stabilized by E. The irreducibility of this representation modulo all primes shows that L* may be scaled to be unimodular. We shall construct a group E* of transformations of 248-space; it will be shown by computer that E^ E*. If p eH is any non-central involution, set F= O2 (CE (aj) n O2 (C£(jS)), D= iVE (F). F is elementary abelian of order 25, D/F SL 2 (5) and D does not split over F. U. Dempwolffhas shown, in [2], that if such an extension exists it is unique. Accordingly we term D (which we construct explicitly) the Dempwolff Group. Every faithful representation of D over the complex field has degree^ 248, there being a unique faithful representation (j) of degree 248, to which the 248-representation of E must restrict. Thompson has constructed a decomposition of the complex Lie-algebra