A Class Of Solvable Transitive Automorphism Group

After the classification of flag-transitive linear spaces, attention has now turned to the flag-transitive automorphism groups. This thesis is a part of this project.The research on t - designs is very important in combinations and designs. Few years ago, Buekenhout, Delandtsheer, Doyen, Kleidman,L-iebeck and Sax effectively classified all regular linear spaces with an aut-omorphism group transitive on flags. Recently, A. R. Camina provides a scheme to classify the block-transitive designs. It says that if G is block-transitive and point-primitive, then the socle of G is either elementary ab-elian groups or non-abelian simple groups. So we can discuss the structure of G by using the classification of finite simple groups theorem.This thesis consists of three chapters.In chapter 1, we will give some introduction about the history and current research situation of the group theory and design (linear spaces) theory. Then we can realize the situation about the development about this research fields.In chapter 2, we will introduce the elementary concepts that will be used in this thesis. Then we can construct the basic knowledge system of this thesis.In chapter 3, we have given the characters of finite linear space. Using the group theory, we discussed the soluble block-transitive automorphism group of 2 -(v,12,1) designs, and get the main theorem as follows:Main Theorem :LetG be a soluble block-transitive automorphism group of 2-(v,12,1) Designs, then G is point-primitive and one of the following holds:(1) v =3¹⁰ and G = Z₃¹⁰:H, where H is a soluble and irreducible subgroup of GL(10,3).(2) v = pⁿ ,where p≠2 is a prime and n a positive integer. When p = 3, then 10|n , G≤A(?)L(1,3ⁿ) and 3ⁿ=1(mod44) .when p≥5 ,then G≤A(?)L(1,pⁿ) and pⁿ =l(mod132).