Almost Simple Group With Si Tannuo 4 - Design

In the research about the group-theory and design-theory, taking useof the automorphism groups of designs can help us to find some newdesigns; Conversely, studying the groups which are regarded as theautomorphism groups of a design can make us understand clearly thestructure of some groups. Hence, the work that the consequence of theclassification of finit simple groups is used to solve design problem hasbecome a hot topic:This thesis is devoted to the research about the case that theautomorphism groups of Steiner 4-Design is flag-transitive, and it wouldbe the most important parts of the work that we finish the classification ofall flag-trasitive Steiner 4-Designs.Firstly, we give some introduction about the history and currentresearch situation of the group-theory and design-theory, and then weintroduce the elementary concepts about it, all of them form the basicknowledge system of this thesis.Secondly, in chapter 3 and in chapter 4, after introducing somecharacterizations of almost simple type groups and some importantconcepts and lemmas, we prove two main theorem about almost simpletype groups acting on a non-trivial Steiner 4-design respectively.Main theorem 1 Let D=(X,B,I) be a design, and G be aflag-transitive automorphism group of D, if G is Almost SimpleType, then Soc(G) is not isomorphic to Suzuki group Sz(q) andRee group Re(q).Main theorem 2 Let D=(X,B,I) be a non-trivial Steiner4-design, andG be a flag-transitive automorphism group of D , if Gis Almost Simple Type, thenâ…°) Soc(G) is not isomorphic to the following groups:(1). N = PSL(2,11), v=1l;(2). N= PSL(2,8), v=8;(3). N=M_v,v=12,22,24;(4). N=M₁₁, v=12; (5). N=A₇, v=15;(6). HS, v=176;(7). CO₃, v=276;(8). N=A_v, v≥5.â…±) If N=M₁₁, v=11, then D is a Steiner 4-(11,5,1) design, andG(?)M₁₁.â…²) If N=M₂₃, v=23, then D is a Steiner 4-(23,7,1) design,and G(?)M₂₃.