Several Types Of Almost Simple Groups Of 2 - (v, K, 1) Design,

The classification of groups and t-designs, including 2-(v,k,1) designs, has been attracted by experts and scholars all around the world who have done extensive research on it since the classification of finite simple groups was finished in 1980s. This essay has done something on the classification of 2-(v,k,1) designs and obtained some creative conclusions.The classification of designs admitting point 2-transitive automorphism groups was solved by Kantor using the theory of finite simple classification. The classification of 2-(v,k,1) designs admitting flag-transitive automorphism groups was completed by a team, consisted of six mathematicians, in 1990. The questions on classification of t-designs adimitting weaker conditions were proposed one after another. This essay considers the question proposed by Preager. In addition, we discuss a specific condition, that is T≌PSL(2,qⁿ).In chapter 1, we give some introduction about the history and current research situation of the group theory and design (linear spaces) theory. Then we can realize of the development situation about this research fields and the relations between the group theory and design (linear spaces) theory.In chapter 2, we introduce the elementary concepts and conclusions that will be used in this thesis. Then we can construct the basic theory system of this thesis.Chapter 3 is the main part of this article. First, we introduce background of the question we will discuss. Then we prove the following theory:Theorem 1: Let S = (P,L) be a finite linear space and G be an almost simple group with a normal subgroup T satisfying G≤Aut(S),where T is a non-Abelian simple group. Suppose that S is not a projective plane. IfG acts line-transitively on S, then T acts point transitively on S.Theorem 2: Let T≌PSL(2,q)≤G≤Aut(T)and G act line transitively on 2-(v,k,l) design S , where S is not a projective plane. Then the following holds:(1) If T acts line transitively on S , then S is isomorphic to Witt-Bose-Shrikhande plane.(2) If T donesn't act line transitively onS, then T acts point-transitive on S. Furthermore, when q is even, then we have:(a) G_a=D_2(q+1):(ζ), andζhas order t^d for t an odd prime. What's more G = P(?)Z(2,2^?).(b) T_a2(q+1),G_a=T_a:(ζ) while G_L＞T_a :(ζ^t),andζhas ordert^d for t an odd prime. What's more G = P(?)L(2,2^?).