In this paper,the author discusses the order of an element of periodic groups and free groups,studies the order of an element of the additive group of a ring and establishes the concept of the characteristic number of a ring,and gives two popularizations of the order of an element of groups:the order of an element to a subgroup and the order of an element of semigroups.

Some problems on the order of an element of groups;

Element order is an important quantity in finite groups,which influences the integral property of finite groups profoundly.

To investigate how element orders influence the structure of a finite group is an important subject in group theory.

This thesis mainly considers how the arithmetical conditions of conjugacy classes and element orders of a finite group influence its structure respectively.