Finite Meta-primitive Permutation Groups

In this thesis, we mainly study the meta-primitive permutation groups in per-mutation group theory. Moreover, we classify primitive permutation groups with fourth-power-free orders, and give some properties of the imprimitive permutation groups with rank4.A permutation group G≤Sym(Ω) is called meta-primitive if for each non-trivial G-invariant partition?, the induced permutation group G? is primitive. Primitive permutation groups can be viewed as trivial examples of meta-primitive permutation groups, that is, meta-primitive permutation groups is a natural gen-eralization of primitive permutation groups. Meta-primitive permutation groups involve many important families of permutation groups, such as imprimitive per-mutation groups of rank3, bi-primitive permutation groups, and transitive per-mutation groups of degree pq with p, q distinct primes, etc. Moreover, it is easily shown that each imprimitive permutation group has a quotient action which is meta-primitive. Depending on these observations, classifying meta-primitive per-mutation groups is thus a very important and interesting topic in the permutation group theory.It is well known that there is a famous O'Nan-Scott theorem for primitive permutation groups. In1993, this theorem is developed on quasiprimitive per-mutation groups by C.E.Praeger. By analyzing the structures and the actions of their socles, the O'Nan-Scott's theorem divides primitive permutation groups into eight types. The theorem has been the most important and efficient tool in the study of permutation groups and relative problems. In the present thesis, by studying the minimal normal subgroups, we obtain a theorem in meta-primitive permutation groups, which is similar to the O'Nan-Scott Theorem of primitive permutation groups, ie. Our theorem is a generalization of the O'Nan-Scott the-orem of primitive permutation groups. Moreover, some examples of each type are given.Let G be a permutation group on Ω. If there is no prime p such that p4divides|G|, then G is called a fourth-power-free group. Using O'Nan-Scott theorem, we give a complete classification of primitive permutation groups with fourth-power-free orders. As an application of the comes in algebraic graph theory, we have determined, in another paper, all2-arc-transitive basin graphs admitting a fourth-power free automorphism group.Finally, we investigate permutation groups of rank4. Let G be a permutation group on Ω, and let a∈Ω. Each orbit of Gα:={g∈G|αg=α}, the stabilizer of G on α, is called a suborbit of G on Ω, and the total number of G-suborbits is called the rank of G. It is easily shown that:a transitive permutation group is of rank2if and only if it is a2-transitive. Through the research of many famous mathematicians, including M.W.Liebeck, J.Saxl, C.E.Praeger and Cai Heng Li, a classification of transitive permutation group of rank3is finally obtained. Upon these results, we find some new properties on permutation groups of rank4. Trying to give more deep research on permutation groups will be a very important topic for us in the future.