Linear Space Of The Two Types Of Automorphism Group

After the classification of flag-transitive linear spaces, attention hasnow turned to classification of line-transitive linear spaces. This thesis isa part of this project.Let G be an automorphisms group of a linear space. We know thatthe study of linetransitive finite linear spaces can be reduced to threecases, distinguishable by properties of the action on the point-set: that inwhich G is of affine type in the sense that it has an elementary abeliantransitive normal subgroup; that in which G has an intransitive minimalnormal subgroup; and that in which G is almost simple, in the sense thatthere is a simple transitive normal subgroup T in G whose centraliseris trivial, so that T≤G≤Aut(G). A. R. Camina, C. E. Praeger, P. M.Neumann, F. Spiezia considered the cases where T is isomorphic to oneof 26 sporadic simple groups or an alternating group. In this thesis, weconsider the last case. Namely, there exits a T such that T≤G≤Aut(G) andG acts as line-transitive on finite linear spaces, where T is non-abeliansimple. The main work of this thesis is as follow:1,We consider the cases where the socle of G has Lie rank 1,especially, he socle of G such that Soc(G)(?)²G₂(q), we come to theconclusion that T acts line-transitively on finite linear spaces.2,We take some useful consider on Lie type simple with Lie rank 2,mainly, we study the cases where Sot(G) is isomorphic to G₂(q). In thistime, we also conclude that T acts line-transitively on finite linearspaces.This thesis consists of three chapters.In chapter 1, we will give some introduction about the history andcurrent research situation of the group theory and design (linear spaces)theory. Then we can realize the situation about the development aboutthis research fields.In chapter 2, we will introduce the elementary concepts that will beused in this thesis. Then we can construct the basic knowledge system ofthis thesis.In chapter 3, we will consider almost simple group acting line-transitively on finite linear spaces. We will get the following maintheorems:Main Theorem 1: Let G be an almost simple group actingline-transitively on finite linear space S. If Soc(G)=²G₂(q), then Soc(G)is also line-transitive and S is a Ree unitary space, where Soc(G)donotes the socle of G, q=3²ⁿ⁺¹ and n≥1.Main Theorem 2: Let G be an almost simple group actingline-transitively on finite linear space S=(Ω, B). If Sot(G) is isomorphicto G₂(q), where q =p^αandα＞1. If 6 doesn't divideα, then Soc(G)is line transitive.