Almost Single-group Design, And Symmetry

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. Eugenia and his students have discussed the t-(v,k,λ) designs. They got that the flag-transitive automorphism groups of a ^t- design must be either affined or almost simple whenλ≤3. They have completed the main work of 2-(v,k,2) designs. Based on their researches, we will consider the 2 - (v, k, 3) symmetric designs.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, the characters of symmetric designs have been mentioned. We have also given the structures and characters of some almost simple groups, discussed the flag-transitive automorphism groups. We will get the main theorem as follows:Main Theorem Let G be a primitive, flag-transitive automorphism group of a 2-(v,k,3) symmetric design.(i)If the socle of G is isormorphic to the sporadic groups, then G (?) M₂₃, G_x(?)L₃(4):2 or G_x(?)2⁴:A₇;(ii)If the socle T of G is isormorphic to ²G₂(q) group, ²F₄(q²) group or G₂ (q) group, then T_x must be their subfield groups.(iii) The socle of G can't be S_z(q) group.