Applications Of Finite Groups In Some Combinatorial Structures

The finite groups have great applications in the study of certain combinatorial structures, especially in the field of combinatorial de-signs and graph theory. The construction of simple t-designs with certain transitive properties, for example, flag or block-transitivity, is a hot topic with great theoretic significance and application back-ground in the field of combinatorial design theory. Currently, the re-sults of block-transitive2-designs are abundant. On the other hand, the studies of t-designs for t≥3are much fewer. Thus, for the case t≥3, constructing block-transitive t-designs by studying their auto-morphism groups becomes an important topic and difficult problem. In the field of graph theory, determining whether two given graphs are isomorphic is fundamental for understanding the graphs and for deter-mining isomorphic classes of graphs. For vertex-transitive graphs, one would expect to determine their isomorphisms by a vertex-transitive automorphism group, that is to say, to determine the isomorphism be-tween two G-vertex-transitive graphs by the information of the group G. For Cayley graphs, such problem has been extensively studied over the past decades. Since lots of vertex-transitive graphs are not Cayley graphs, it is a natural next step to extend the study for Cayley graphs to vertex-transitive graphs.The thesis is divided into seven chapters. We mainly study the existence of block-transitive4-designs and6-designs in combinato-rial design theory and the isomorphism problems of vertex-transitive graphs.The first chapter is devoted to surveying research background, results, methods in this area.In the second chapter, we collect some notations and definitions, and quote some preliminary results, which will be used in this thesis. We give those related to abstract group theory, permutation groups, combinatorial designs and graph theory in separate sections.In the third chapter, we study the existence of block-transitive4-(q+1,6, λ) designs admitting the projective special linear group PSL(2, q) or projective general linear group PGL(2, q) as its automor-phism groups. Moreover, we construct some brand new4-(q+1,6,λ) designs with given parameters.In the fourth chapter, we determine the orbits and orbit sizes under the action of PSL(2,q) on the5-subsets of the projective line GF(q) U{∞} for q=1(mod4). Moreover, we construct some simple3-designs with given parameters admitting PSL(2, q) as the automor-phism group.In the fifth chapter, we attack the famous Cameron-Praeger con-jecture. Using the sufficient and necessary conditions for the existence of block-transitive designs and the classification of3-homogeneous groups, we proved that the Cameron-Praeger conjecture is true for k<100.In the sixth chapter, we discuss the isomorphism problem of vertex-transitive graphs, similar to the study for Cay ley graphs. In this chapter, we give the necessary and sufficient conditions for a vertex-transitive graph to be a GI-graph. Moreover, for several fami-lies of finite simple groups, we prove that the corresponding connected vertex-transitive cubic graphs are Gi-graphs.In the seventh chapter, we construct a family of half-transitive graphs, which is generated from the Johnson graph. Further, this family of graphs contains infinite many Cayley graphs and non-Cayley graphs.