3 - Homogeneous Permutation Group And 7 - Design

In1993, Peter J. Cameron and Cheryl E. Praeger conjectured that there are no nontrivial block-transitive6-designs. According to the conjecture, Michael Huber completed the proof of the nonexistence of the block-transitive Steiner6-designs (except possibly when the group is G=PГL(2,pe) with p=2,3, and here e is an odd prime power), and the block-transitive Steiner7-designs. Mainly based on the results and experience of Michael Huber, this paper further classified and discussed the existence of the7-(v,k,λ) designs (when2≤λ≤5ork≤100).This thesis consists of three chapters.In chapter1, we give some introduction about the history and current research situation of the main content of the thesis, and introduced the main job of the paper.In chapter2, we mainly introduced some related basis knowledge of groups and combinational designs in the thesis, and constructed the basic theory system of the paper.In chapter3, based on the results and experience of Michael Huber, we discussed the existence of the7-(v,k,λ) designs (2≤λ≤5), and obtained main theorem as follows:Main Theorem1:Let D=(X,Ð') be a non-trivial7-(v,k,λ) design, for2≤λ≤5. Then G≤Aut(D) can not act block-transitively on D. Main Theorem2:Let D=(X,Ð') be a non-trivial7-(v,k,λ) design,for k≤100.Then G≤Aut(D) can not act block-transitively on D.