The Finite Groups Having Elements Of Maximal Order 2pq~2

With the development of group theory,people know that group theory is a fundamental part of algebra gradually,group theory in math and abstract algebra has basic important position. In the base of group, we can add a new operation and justice to form many algebraic structures, such as the ring, domain and mould and so on. The concept of the group is not only have broad and important application in each of mathematics branch, and the method of group theory research have also use in physics and chemistry study, such as crystal structure, hydrogen atom structure and modeling. So group theory and related group theory have lots of application in physics and in chemistry, other basic science and applied science.In the knowledge of group theory, the solution of group has an important meaning, we can get some conclusions about nilpotent groups, such as"Studies on Nilpotent and Soluble Groups"in Ketta Balake's dissertation. Besides, from its solution of group, we can get the order of Fitting subgroup of limited solvable group and the impact of the distribution state of Zero point in the features table of limited soluble groups to the structure of Group and other aspects.This paper using knowledge and related theory of mormal subgroup, quotient group, Sylow theorem, studies the influence of the number of elements having the highest order to the structure of group. Firstly, we discussed the number of elements having the highest order how to effect the solvability of the finite groups,and then we get it how to effect the structure of the finite groups.There are the following conclusions mainly:This paper discusses the solvability of the finite groups where the number of elements of the maximal order is 2pq~2. Then we get that Thomopson question holds under certain condition if |M(G)| = 2pq~2.