Study On Groups With Some Local Properties

In this thesis, we mainly study the suborbit of primitive permutation group and the derived length of solvable group.Research on the suborbit of primitive group originated from a beautiful conclusion: the primitive group with a suborbit of length 2 must be isomorphic to D2p (p is an odd prime). Since then, study the suborbit of primitive group has attracted wide attention and become a very active research topic. Especially, classify the primitive group with a suborbit of length p is of vital importance. When p> 2, the situation becomes much more complex. Wang Jie and many scholars studied the primitive permutation group which had a suborbit of length p<5, and completely classified the primitive group which had a suborbit of length 3 and 4. For p= 5, professor Wang Jie classified the primitive group with subconstituent GαΔ(α) was solvable and GαΔ(α)> A5 was non-faithful. On the basis of his research, we classify the primitive group with subconstituent GαΔ(α)> A5 and is faithful in the third chapter of this paper. Together with the results of W.Quirin and Wang Jie, we finally complete the classification of all primitive groups with a suborbit of length 5. Furthermore, in the fourth chapter of this paper we classify the primitive permutation group which has a suborbit of length 7 and the subconstituent GαΔ(α) is solvable.In my study, we make use of the knowledge of permutation group theory, represen-tation theory, Aschbacher theorem and the structure of maximal subgroups of typical group, firstly attribute G to affine and almost simple primitive. For the affine primitive group, the equivalent of the irreducible representation according to the representation theory; for the almost simple group, we make use of the structure of maximal subgroups of simple group and analyze whether their maximal subgroup is isomorphic to A5 or S5, furthermore determine whether there is a suborbit meeting the conditions.The second part of my paper is to research on the solvable group. After the comple-tion of finite simple group classification theorem, research on solvable group have become a hot spot. For example, research derived length of solvable group gets spotlight atten-tion. Professor Li Shirong got an upper bound of derived length d(G) of solvable group, that is d(G)≤2(δ(G)-1)1/2+1, here denote by δ(G) the number of conjugacy classes of non-cyclic subgroup. Along with his work, in the fifth chapter of this paper we also get an upper bound of derived length, that is d(G)≤2(δ*(G)-λ(G)-1)1/2+1, here λ(G) is the number of i such that G(i)/G(i+1) is a non-cyclic 2-group and denote by δ(G)* the number of conjugacy classes of non-abelian subgroup, which greatly improves the result of the above.Moreover, in the final part of this paper, we consider the primitive group acting on 2-arc transitive Cayley graph T=Cay(G, S). We use the classifications of primitive permutation group including a regular subgroup and 2-transitive group, combine with the necessary and sufficient conditions for 2-arc transitive graph, we get a basic depiction of (X,2)-arc transitive graph Γ if G≤X< Aut(Γ) and X acting on the vertices G is almost simple primitive group.