The Influence Of The Orders And The Number Of Subgroups Of Finite Groups On Groups

We all know that it is necessary to know a lot of quantity properties of finite groups in the research of finite groups theory.For example,the orders of groups;the orders of elements of groups;the number of same order elements;the number of conjugate classes of subgroups;the number of Sylow p-subgroups.These quantity properties are the important contents of the finite groups theory.Conversely,if we know some quantity properties of finite groups,can we get their some properties? Facts show that it indeed can obtain good properties of finite groups from some simple quantity properties.For example,if a finite group has only one Sylow p-subgroup,then this Sylow p-subgroup is normal and a characteristic subgroup;odd order group is solvable;Ï€_e(G) represents the set of elements' order in G,IfÏ€_e(G)={1,2,3,5},then G is isomorphic to A₅.The order of finite group is also an important quantity property.Chapter two of this paper mainly discusses the influence of the set of indexes of subgroups on finite groups, and gets that if the sets of indexes of subgroups of two groups are equal,where,one group is solvable,then the other is also solvable,and their orders are equal.Moreover, this thesis also studies the continuity of the numbers in the set of indexes of subgroups of finite groups.If the sets of indexes of subgroups of two finite groups are equal,then the sets of oders of subgroups of this two groups are also equal.In the third chapter,the author discusses the influence of the set of orders of subgroups on simple groups,and characterizes some simple groups by the set of orders of subgroups,from which we can get that if the sets of orders of subgroups of two simple groups are equal,then these two simple groups are isomorphic.The number of subgroups has much influence on the structures of finite groups. Many researchers on the group theory have studied a lot.For example,G.A.Miller has given all finite groups whose subgroups number is eleven or twelve,and get the method that can calculate the number of subgroups of Abelian groups.In the forth chapter, the author gives a new characterization of some finite groups by the number of their subgroups,that is when |G|=2p,p is odd prime;2²p,p is a odd prime;3p²(p＞3,p is a prime,3 doesn't divide p-1),then G is only determined by m.