Primitive permutation groups with soluble stabilizers and applications to graphs

In the past 60 years interactions between group theory and the theory of graphs have greatly stimulated the development of each other, especially the theory of symmetric graphs (or more generally vertex-transitive graphs) has almost developed in parallel with the theory of permutation groups. The aim of this thesis is to make an e ort, both in terms of pure research and broader value, to solve some challenging problems in these two elds. First we considered the problem of classifying nite primitive permutation groups with soluble stabilizers. This problem has a very long history. By the O'Nan- Scott Theorem, such groups are of type a ne, almost simple, and product action. We reduced the product action type to the almost simple type, and for the latter, a complete classi cation was given, which forms the main result of the thesis. The main result was then used to solve some problems in algebraic graph theory. The rst application of which is to classify edge-primitive s-arc-transitive graphs for s 4. After undertaking a general study on the local structures of 2-path-transitive graphs, we presented the second application by classifying nite vertex-primitive and vertex-biprimitive 2-path-transitive graphs. The result then helps us to be able to construct some new half-transitive graphs. Another application of the main result is that a complete classi cation of nite vertexbiprimitive edge-transitive tetravalent graphs is given (recall that for the cubic case, the classi cation was given by Ivanov and Io nova in a highly cited article in 1985).