The simple group of order 604,800
The simple group of order 604,800
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简单组 604,800
DOI：
10.1016/0021-8693(68)90014-8
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发表时间：
1968
影响因子：
0.9
通讯作者：
D. B. Wales
中科院分区：
文献类型：
--
作者：
M. Hall;D. B. Wales
最近2。Janko指出，如果单群G在2-Sylow子群的中心有一个对合的正化器，并且G有两个共轭对合类，则其阶数为604,800，其特征表为他所描述的。本文独立地证明了一个阶为604800的简单群的存在唯一性。在604800阶的单群G存在的假设下，得到了它的特征表。这项工作在第2-14节中给出。在第15节中，我们证明了G必须包含6048阶的子群。在第16节和第17节中，我们找到了G在H的100个集上的排列表示，并且证明了这种表示是唯一的。这也证明了这个群是简单的。排列表示的正确性在剑桥大学的Titan计算机上得到了Peter Swinnerton-Dyer的帮助。从而建立了G的存在唯一性。在第18节中，Janko对G的描述是成立的。第19节给出了G的2阶外自同构，填充了特征表的无理数部分，证明了G有指标为100、280、315的子群，没有其他更小指标的子群。我们使用关于元素和子群的中心化器和规范化器的标准符号。因此，C (H)和N (H)是子集H的中心化器和规范化器，C (H)和N (H)是k中H的中心化器和规范化器，C (H) N H表示为Z (H)。集合S生成的群记为(S)。如果L ‘是G中的元素，则C (n)和n（17）表示C （(n)）和n （（L ’））。如果S是一个子集1，s1表示S的基数。Wesetg=/GI= 604,800。
Recently 2. Janko indicated that if a simple group G had a certain centralizer for an involution in the center of a 2-Sylow subgroup and if G had two conjugate classes of involutions, its order would be 604,800 and its character table would be as he described. The existence and uniqueness of a simple group of order 604,800 is proved here independently. Under the assumption of the existence of a simple group G of order 604,800, its character table is found. This work is given in Sections 2-14 inclusive. In Section 15 it is shown that G must necessarily contain a subgroup of order 6048. In Sections 16 and 17 the permutation representation of G on the 100 cosets of H is found, and this representation is shown to be unique. It is also shown that this group is simple. The correctness of the permutation representation was verified with the help of Peter Swinnerton-Dyer on the Titan computer at Cambridge University. Thus the existence and uniqueness of G is established. In Section 18 Janko’s characterization of G is shown to hold. Section 19 gives the outer automorphism of G of order 2, fills in the irrational part of the character table and shows that G has subgroups of indices 100, 280, 315, and no other subgroups of smaller indices. We use standard notation regarding centralizers and normalizers of elements and subgroups. Thus C (H) and N (H) are the centralizers and normalizers of subsets H. Also C,(H) and N,(H) are the centralizers and normalizers of H in K. Also C (H) n H is denoted as Z (H). The group generated by a set S is written (S). If L’is an element in G, C (n) and N (17) mean C ((n)) and N ((L’)). If S is a subset 1 S 1 denotes the cardinality of S. Wesetg=/GI= 604,800.