Transitive T-designs With Large T

Abstract:Cameron and Praeger proved a very meaningful theorem: When t≥8, there are no nontrivial block-transitive t-designs, and when t≥7, there are no nontrivial flag-transitive t-designs. Michael Huber completed the proof of the nonexistence of the flag-transitive Steiner6-designs and block-transitive Steiner7-designs. To continue the work of experts, this paper mainly discuss the existence of nontrivial flag-transitive6-(v,k,λ) designs and nontrivial block-transitive7-(v,k,λ) designs.This paper consists of three chapters.In chapter1, we give some introduction about the background and current research situation of the subject. And summarize the main work of this thesis briefly.In chapter2, we introduce some basic theoretical knowledge of groups and combinational designs. Especially some of the theorems and lemmas which will be very important in our proof of the main theorem.In chapter3, we discuss the existence of nontrivial flag-transitive6-(v, k, λ) designs and nontrivial block-transitive7-(v, k,λ) designs. We obtained two main theorems as follows:Main Theorem1:Let D=(X,B) be a non-trivial and simple6-(v,k,λ) designs. If G≤Aut(D) acts flag-transitive on D, then λ>20.Main Theorem2:When6≤λ≤10, there are no nontrivial and simple7-(v, k,λ) designs which are block-transitive.