Some topics in the theory of finite groups
Some topics in the theory of finite groups
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有限群理论中的一些主题
DOI：
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发表时间：
1971
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通讯作者：
D. E. Taylor
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作者：
D. E. Taylor
常规 2-图由集合 andOmega 组成；与 andOmega 的三元素子集的（非空）集合 t 一起；使得 andOmega 的任意二元素子集；包含在与 t 相同数量的元素中，即 andOmega 的任何四元素子集；包含 t 的偶数个元素，而不是 andOmega 的每个三元素子集；位于 t 中。这些对象是由 G.andnbsp;Higman 引入的，他使用具有 276 个点的正则 2-图来为康威的零星单群 C 3 的双重传递表示提供组合设置。本论文表明，正则2-图与强图的等价类（由J.J.andnbsp；Seidel 定义）一一对应。此外，对于正则 2-图的每个点，有一种自然的方法可以在其余点上定义强正则图。这些图形表示用于获得对常规 2-图的结构和参数的限制。通过强图，也可以将正则 2-图表示为欧几里德空间中等角线的配置。相反，本文获得的关于正则2图的结果扩展了J.J.和Seidel在等角线上的结果。构造常规 2-图，其允许 PSL(2,q) 、qandequiv;l (mod 4)、Sp(2m,2) 在两种双重传递表示中； PSU(3,q 2 ), q 奇数；所有 Ree 型群连同 2 G 2 (3) = Aut(PSL(2,8))；零星的简单群C 3 和HiS；群 V.Sp(2m,2) 是 2m 维向量空间在域 GF(2) 上平移的群 V 与 Sp(2m,2) 的半直积。通过研究与双传递表示相关的单项式表示的中心环，表明（除了某些具有正则正规子群的群可能例外）上述群是唯一已知的可以充当正则 2-图自同构的双传递群的群。
A regular 2-graph consists of a set andOmega; together with a (non-empty) set t of three-element subsets of andOmega; such that any two-element subset of andOmega; is contained in the same number of elements of t , any four-element subset of andOmega; contains an even number of elements of t and not every three-element subset of andOmega; is in t . These objects were introduced by G.andnbsp;Higman who used a regular 2-graph with 276 points to provide a combinatorial setting for the doubly transitive representation of Conway's sporadic simple group C 3 . In this thesis it is shown that regular 2-graphs are in one-one correspondence with equivalence classes of strong graphs (as defined by J.J.andnbsp;Seidel). Moreover, for each point of a regular 2-graph there is a natural way of defining a strongly regular graph on the remaining points. These graphical representations are used to obtain restrictions on the structure and on the parameters of a regular 2-graph. It is also possible, via the strong graphs, to represent a regular 2-graph as a configuration of equiangular lines in Euclidean space. Conversely, results about regular 2-graphs obtained in this thesis extend the results of J.J.andnbsp;Seidel on equiangular lines. Regular 2-graphs are constructed which admit the PSL(2,q) , qandequiv;l (mod 4), Sp(2m,2), in both doubly transitive representations; PSU(3,q 2 ), q odd; all groups of Ree type together with 2 G 2 (3) = Aut(PSL(2,8)); the sporadic simple groups C 3 and HiS; the group V.Sp(2m,2) which is the semi-direct product of the group V of translations of a vector space of dimension 2m over the field GF(2) by Sp(2m,2). By studying the centraliser ring of a monomial representation associated with the doubly transitive representation it is shown that (with the possible exception of some groups with a regular normal subgroup) the above groups are the only known groups which can act as doubly transitive groups of automorphisms of a regular 2-graph.