本文關(guān)鍵詞： 對稱設(shè)計(jì) 非對稱設(shè)計(jì) 旗傳遞 點(diǎn)本原 自同構(gòu)群　出處：《華南理工大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：一個(gè)2-(v,k,λ)設(shè)計(jì)D是一個(gè)關(guān)聯(lián)結(jié)構(gòu)(P,B),其中P是v個(gè)點(diǎn)的集合,召是P的k-元子集的集合,召中元素被稱為區(qū)組,每個(gè)區(qū)組至少與兩個(gè)點(diǎn)關(guān)聯(lián),且滿足P的任意2-元子集恰好與λ個(gè)區(qū)組關(guān)聯(lián).設(shè)計(jì)D的自同構(gòu)群是保持召不變的P上的置換群.對設(shè)計(jì)及其自同構(gòu)群的研究是群論與組合論的一個(gè)重要課題,其內(nèi)在聯(lián)系主要通過自同構(gòu)群具有群的某些性質(zhì)來體現(xiàn)的.一方面,具有傳遞性,本原性等性質(zhì)的自同構(gòu)群可以幫助我們發(fā)現(xiàn)新的設(shè)計(jì)和分類設(shè)計(jì);另一方面,研究設(shè)計(jì)的結(jié)構(gòu)及分類能夠使我們更直觀的了解群的結(jié)構(gòu)及性質(zhì).本文討論旗傳遞設(shè)計(jì)的分類,試圖通過對特殊情形的研究來揭示一般的規(guī)律.旗傳遞2-(v,k,λ)對稱設(shè)計(jì)的分類問題是群與組合設(shè)計(jì)相互作用的一個(gè)典型問題,尤其是當(dāng)λ較小的對稱設(shè)計(jì)被很多學(xué)者研究,并取得了豐碩的成果.對稱設(shè)計(jì)的研究由來已久,本文將對稱設(shè)計(jì)的研究方法應(yīng)用到非對稱設(shè)計(jì)上,首先對非對稱的2-(v,k,λ)設(shè)計(jì)進(jìn)行討論,得到一類旗傳遞非對稱設(shè)計(jì)的分類.然后我們放大參數(shù)λ的范圍,對自同構(gòu)群的基柱是交錯(cuò)群的對稱設(shè)計(jì)進(jìn)行討論.其次,我們研究了λ=4時(shí)自同構(gòu)群的基柱是例外李型單群的旗傳遞對稱設(shè)計(jì),并且接著討論了當(dāng)λ任意時(shí),此類對稱設(shè)計(jì)的分類問題.最后,我們利用單群的大子群的分類,討論了當(dāng)旗傳遞點(diǎn)本原自同構(gòu)群G同構(gòu)于任意一個(gè)例外李型單群時(shí),其大子群的分類問題.本文的主要結(jié)果如下定理3.0.1.設(shè)D是一個(gè)非對稱的2-(v,k,λ)設(shè)計(jì),滿足條件(r,λ)=1,其中r表示過一點(diǎn)區(qū)組的數(shù)目.若G≤Aut(D)是旗傳遞的,且基柱Soc(G)= An,那么在同構(gòu)意義下,非對稱設(shè)計(jì)D和自同構(gòu)群G是下列之一：(ⅰ)D是唯一的2-(15,3,1)設(shè)計(jì),且G=A7或A8；(ⅱ)D是唯一的2-(6,3,2)設(shè)計(jì),且G=A5(ⅲ)D是唯一的2-(10,6,5)設(shè)計(jì),且G=A6或S6.定理4.0.1.設(shè)D是一個(gè)2-(v,k,λ)對稱設(shè)計(jì),滿足(r,λ)2≤λ,其中r表示過一點(diǎn)的區(qū)組數(shù)目.若自同構(gòu)群G≤Aut(D)是旗傳遞的,且基柱Soc(G)= An(n≥5),則D是一個(gè)2-(15,7,3)設(shè)計(jì),且G=A7或A8.定理5.0.1.設(shè)D是一個(gè)非平凡的2-(v,k,4)對稱設(shè)計(jì),G Aut(D)是旗傳遞點(diǎn)本原的,那么Soc(G)不能是例外李型單群.定理6.0.1.設(shè)D=(P,B)是一個(gè)旗傳遞點(diǎn)本原的2-(v,k,λ)對稱設(shè)計(jì),G≤Aut(D)是幾乎單型本原群,即X≤G≤Aut(X),X是一個(gè)非交換單群,那么X不能群Sz(g),~2G_2(g),~2F_4(g)或者~3D_4(q).定理7.0.1.設(shè)D是一個(gè)2-(v,k,λ)對稱設(shè)計(jì),且G≤Aut(D)是旗傳遞點(diǎn)本原的,若G同構(gòu)于例外李型單群,則下列之一成立：
[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，

本文編號：1534028