Order Of Elements Of Groups And Groups

The aim of the study of group theory is to investigate the structure of all groups, It is an important way to study the structure of groups by means of orders of elements of groups.In this paper,in view of orders of elements, we study the order of the product of two elements and the method to the maximal order of elements if it exists, Moreover,we study related properties and structure of groups, and obtain the description of finite groups of several types. The main results are as follow.Theorem 1, Let a andb be two elements of a finite groupG , suppose that o(a)=m,o(b)=n,and ab=ba, put(m,n)=d,m=m′d,n=n′dtheno(ab)=m′n′d₁ with d₁ |d .Theorem 2, Let a and b be two elements of a finite group H , if|a|=|b|=manda=b^p for some p∈NTheorem 3, Let a and b be two elements of a finite groupG ,if |a|=|b|=m,and there exist p and q in N ,such that a^p = b^q,thenTheorem 4, If there exists an element of the greatest order in an abelian group, then this greatest order can be obtained as follows:1) If the cyclic group of order, the order of elements is the most advanced stage.2) For the average exchange group, be obtained as follows:Let the order of elements is the order is (we may assume), there is order in the group is the element, and we know that the group there must be elements of order, and we know that the group there must be elements of order,This continued to do so, you can find the most advanced elements of the band.Theorem 5, Let H be a group of order 2qp , where q and p are different odd prime numbers and q < p, then H≌G_i if and only if1)Ï€_e(H)=Ï€_e(G_i);2)|H|=G_i where i=1,2,3,4,5,6.Theorem 6, LetG and H be groups of order 2 3p where p is an odd prime number, then H≌G_i if and only if:1)Ï€_e(H)=Ï€_e(G_i);2)|H|=G_i,where i = 1,8,14,15,16,18.In addition, we also generalize a result of [ 23] .