射影线性群区传递作用于5-（q+1，7，λ）设计

【作者】 杨冠；

【导师】 刘伟俊；

【作者基本信息】 中南大学 ， 基础数学， 2011， 硕士

【摘要】 群在抽象代数中具有基本的重要地位,许多代数结构,包括环、域和模等可以看作是在群的基础上添加新的运算和公理而形成的。群的概念在数学的许多分支都有出现,而且群论的研究方法也对抽象代数的其它分支有重要影响。本文旨在讨论特殊射影线性群PSL(2,q)和一般射影线性群PGL(2,q)区传递作用下的5-(q+1,7,λ)设计的存在性问题。本文共分为三个部分：第一部分,我们对群与组合设计的历史背景和研究现状进行了综述,并介绍了本文所做的主要研究内容.第二部分,我们给出了群论与组合设计的一些基本概念,为后面章节的讨论和文章的构架打下基础.第三部分,主要对特殊射影线性群PSL(2,q)和一般射影线性群PGL(2,q)区传递作用下5-(q+1,7,λ)设计的存在性问题进行讨论,并在设计存在的情况下,进行设计的构造。我们有以下主要结论：定理1：设D=(X,B“)是一个5-(q+1,7,λ)设计,PSL(2,q)区传递作用在X上,X=GF(q)∪{∞},则当q=23时,存在两个不同构的5-(q+1,7,λ)设计,其中λ=3.定理2：设D=(X,BG)是一个5-(q+1,7,λ)设计,PGL(2,q)区传递作用在X上,X=GF(q)∪{∞},则下列情形发生：(1)当q=1 7时,存在唯一的5-(q+1,7,λ)设计,其中λ=3；(2)当q=23时,存在唯一的5-(q+1,7,λ)设计,其中λ=6.更多还原

【Abstract】 Groups take a basic importance in abstract algebra. Many algebraic structures, including rings, fields, and molds that can be seen as the basis of the group to add new operations and axioms formed. The concept of group of has emerged in many branches of mathematics, and the group theory methods are also has an important effect in other branches of abstract algebra.This paper aims at discussing the existence of the 5-(q+1,7,λ) designs admitting the block transitive automorphism groups projective special linear group PSL(2,q) and projective general linear group PGL(2,q). This thesis consists of three departments.In chapter 1, we give some introduction about the history and current research situation of the group theory and design, and we describe the major research by this article.In chapter 2, we introduce the elementary concepts of the group theory and design that will be used in this thesis.In chapter 3, we focus on discussing the existence of the 5-(q+1,7,λ) design admitting the block transitive automorphism groups PSL(2,q) and PGL(2,q).Then we get some designs with the existence of the 5-(q+1,7,λ) design. We have the main theorem as follows: Theorem 1:Let D=(X,BG) is a 5—(q+1,7,λ)design admitting the block transitiVe automorphism groups PSL(2,q),X=GF(q)∪{∞}.Thern q=23, there is two 5—(q+1,7,λ)designs with not automorphism, whereλ=3.Theorem 2:Let D=(X,BG) is a 5—(q+1,7,λ)design admitting the block transitive automorphism groups PGL(2,q),X=GF(q)u{∞).Then the following may happen: (1)q=17,D=(X,BG)is a 5—(18,7,3)design；(2)q=23,D=(X,BG)is a 5—(24,7,6)design.更多还原