Studying Some Group Problems Associated With Conjugacy Classes

This thesis studies several problems concerning conjygacy classes of finite groups, and it includes two parts. The first part focuses on the groups whose order is divisible by every size squared of conjugacy classes. We prove that if a finite group is the group whose order is divisible by every size squared of conjugacy classes, it must not be a simple group and almost simple group; Abelian groups must be the group whose order is divisible by every size squared of conjugacy classes; The Sylow p-subgroups of nilpotent group are all a group whose order is divisible by every size squared of conjugacy classes if and only if nilpotent group is the group whose order is divisible by every size squared of conjugacy classes. Besides, we discuss when a finite p group is the group whose order is divisible by every size squared of conjugacy classes, and prove that a finite p group is the group whose order is divisible by every size squared of conjugacy classes when its power exponent of order is less than or equal to or4. When a p-group whose order is of the power exponent is greater than4, it must not be a p-group of maximal class. Suppose that the power exponent of order of finite p group G equals5or6. If G is of maximal class, G must not be the group whose order is divisible by every size squared of conjugacy classes; Otherwise, it is the group whose order is divisible by every size squared of conjugacy classes. Special p groups, extral special p groups, Dedekind p groups, inner abelian p groups are all the groups whose order is divisible by every size squared of conjugacy classes.The second part discusses the problems regarding the average size of conjugacy classes of finite groups. We prove that a finite group is super-solvable group when its average length of conjugacy classes is9/5; A finite nonabelian group is a solvable group when its average size of conjygacy classes is p3/p2+p-1. Here p is the minimal prime divisor of the order of group.