The Influence Of Order Equation, Type Of Element Order And The Number Of Elements With Maximal Order On Finite Groups

We know that class equation has much influence on the structure of finite groups. For example,class equation can determine the simple group with minimal order,namely A₅.If we put the conjugcy classes with the same order in the same place,we can get the order equation.Obviousely,order equation is a rough division to the groups.Prof. Shi Wujie posed the definition,of order equation at first in[2].The famous question of Thompson is related to order equation.Let G₁ and G₂ are finite groups,if |M_t(G₁)|= [M_t(G₂)|,t = 1,2,3,···,M_t(G)= {x∈G:x^t = 1},then we say G₁ and G₂ are the same type of order.LetG₁ and G₂ are the same type of order,if G₁ is solvable,then is G₂ also sovable? That is the famous question.It is very difficult to solve the open question.The second chapter will charicterize L₂(2^m)according to order equation.If type is the same,order equation is also the same.So the study of this chapter will be advantageous to the solution of the question.Obviously,the number of elements with maximal order is the same,if groups have the same type of order.So the third chapter will be useful to the solution of the question.Some specialists on group theory did much great study on this famous question. The fourth chapter will discuss another question which is also related to the famous question.So the fourth chapter will be helpful to the solution of the question.