2-（v，k，λ）设计的旗传递自同构群

发布时间：2018-02-25 10:20

  本文关键词： 对称设计 非对称设计 旗传递 点本原 自同构群　出处：《华南理工大学》2016年博士论文　论文类型：学位论文

【摘要】：一个2-(v,k,λ)设计D是一个关联结构(P,B),其中P是v个点的集合,召是P的k-元子集的集合,召中元素被称为区组,每个区组至少与两个点关联,且满足P的任意2-元子集恰好与λ个区组关联.设计D的自同构群是保持召不变的P上的置换群.对设计及其自同构群的研究是群论与组合论的一个重要课题,其内在联系主要通过自同构群具有群的某些性质来体现的.一方面,具有传递性,本原性等性质的自同构群可以帮助我们发现新的设计和分类设计;另一方面,研究设计的结构及分类能够使我们更直观的了解群的结构及性质.本文讨论旗传递设计的分类,试图通过对特殊情形的研究来揭示一般的规律.旗传递2-(v,k,λ)对称设计的分类问题是群与组合设计相互作用的一个典型问题,尤其是当λ较小的对称设计被很多学者研究,并取得了丰硕的成果.对称设计的研究由来已久,本文将对称设计的研究方法应用到非对称设计上,首先对非对称的2-(v,k,λ)设计进行讨论,得到一类旗传递非对称设计的分类.然后我们放大参数λ的范围,对自同构群的基柱是交错群的对称设计进行讨论.其次,我们研究了λ=4时自同构群的基柱是例外李型单群的旗传递对称设计,并且接着讨论了当λ任意时,此类对称设计的分类问题.最后,我们利用单群的大子群的分类,讨论了当旗传递点本原自同构群G同构于任意一个例外李型单群时,其大子群的分类问题.本文的主要结果如下定理3.0.1.设D是一个非对称的2-(v,k,λ)设计,满足条件(r,λ)=1,其中r表示过一点区组的数目.若G≤Aut(D)是旗传递的,且基柱Soc(G)= An,那么在同构意义下,非对称设计D和自同构群G是下列之一：(ⅰ)D是唯一的2-(15,3,1)设计,且G=A7或A8；(ⅱ)D是唯一的2-(6,3,2)设计,且G=A5(ⅲ)D是唯一的2-(10,6,5)设计,且G=A6或S6.定理4.0.1.设D是一个2-(v,k,λ)对称设计,满足(r,λ)2≤λ,其中r表示过一点的区组数目.若自同构群G≤Aut(D)是旗传递的,且基柱Soc(G)= An(n≥5),则D是一个2-(15,7,3)设计,且G=A7或A8.定理5.0.1.设D是一个非平凡的2-(v,k,4)对称设计,G Aut(D)是旗传递点本原的,那么Soc(G)不能是例外李型单群.定理6.0.1.设D=(P,B)是一个旗传递点本原的2-(v,k,λ)对称设计,G≤Aut(D)是几乎单型本原群,即X≤G≤Aut(X),X是一个非交换单群,那么X不能群Sz(g),~2G_2(g),~2F_4(g)或者~3D_4(q).定理7.0.1.设D是一个2-(v,k,λ)对称设计,且G≤Aut(D)是旗传递点本原的,若G同构于例外李型单群,则下列之一成立：
[Abstract]:A 2-D) design D is an associated structure, where P is a set of v points, a call is a set of k-element subsets of P, and the call element is called a block, each block is associated with at least two points. The automorphism group of design D is a permutation group on P that remains unchanged. The study of design and automorphism group is an important subject of group theory and combination theory. On the one hand, automorphism groups with transitivity and primitivity can help us to find new designs and classification designs, on the other hand, The structure and classification of design can help us to understand more intuitively the structure and properties of group. This paper discusses the classification of flag transfer design. This paper attempts to reveal the general law by studying the special case. The classification problem of flag transfer 2-v / v / k, 位) symmetric design is a typical problem of the interaction between group and combinatorial design, especially when the symmetric design with smaller 位 is studied by many scholars. In this paper, the research method of symmetric design is applied to asymmetric design. The classification of a class of flag transfer asymmetric designs is obtained. Then we enlarge the range of parameter 位 and discuss the symmetric design that the base column of an automorphism group is a staggered group. In this paper, we study the flag-transitive symmetric design that the base column of 位 = 4:00 automorphism group is an exception lie type simple group, and then discuss the classification problem of such symmetric design when 位 is arbitrary. Finally, we use the classification of the large subgroup of a simple group. This paper discusses the classification of large subgroups of flag transfer point primitive automorphism group G when G is isomorphic to any exceptional lie simple group. The main results of this paper are as follows: theorem 3.0.1. If G 鈮，

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