The Projective Linear Group Transitive Role Of 5 - (q +1,7, ¦Ë) Design,

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.