Fang-li Parameters And Delandtsheer Conjecture

This paper studies the Delandtsheer conjecture. In1988, Delandtsheer gave thefollowing conjecture:Let D be a2-(v, k,1) design, if G is a block-primitive automorphism group of D,then G is also point-primitive.Many scholars have studied Delandtsheer conjecture and proved that it is true un-der some additional conditions. This thesis focuses on the conjecure and proves thatDelandtsheer conjecture holds when Fang-Li parameter (k,v1k1)=(k, r) is at most20.In1993, Fang Weidong and Li Huiling introduced a set of parameters which arecalled Fang-Li parameters, and proved Camina-Gagen theorem again by using these para-menters:Let D be a2-(v, k,1) design, if G is a block-transitive automorphism group of D andk|v, then G is flag-transitive.From this theorem, we know that G is flag-transitive when (k,v1k1)=(k, r)=1.So G is point-primitive. Then Fang Weidong and Li Huiling proved that the conjectureholds when (k, r)≤4. After that, Liu Weijun, Ma Yanbo and Tian Delu, et al. extendedthis bound up to18. This paper proves that the conjecture is true when k(r)=(k, r)=19or20.Main Theorem: Assume that D is a2-(v, k,1) design, if G≤Aut(D) is block-primitive and k(r)=(k, r)=19or20, then G is also point-primitive.The structure of this thesis is as follows:In Chapter1, we introduce the backgrounds and modern developments of Delandt-sheer conjecture and the result of this paper.In Chapter2, we give some preliminary rusults. Firstly, it is on some definitionsand theorems of permutation groups. Secondly, it is about combinatorial designs andtheir automorphism groups. Minewhile, we introduce Fang-Li parameters, Delandtsheer-Doyen parameters and their properties. Finally, we give some important lemmas whichare needed in this thesis.In Chapter3, we prove the main theorem in three steps. Firstly, based on the prop-erties of parameters, we obtain2819-tuples by using computer. Secondly, we eliminate2729-tuples by using the methods of group and design theory. Finally, we eliminate the99-tuples left step by step. Then we complete the proof of the main theorem.