Using the theory of elementary group and character, we get a result to characterize the structure of finite groups when the average length of conjugacy classes of the finite groups is given. This is significant for studying the quantitative property of finite groups. 利用初等群论方法和有限群特征标理论,在共轭类平均长度为某一定数时,获得了对有限群结构的刻划,且对有限群数量性质的研究是有意义的。
The second part : We study the connection between a non-abelian finite p-group G and the number of conjugacy class of it. We also find some more concise relationships between the minor order of a non-abelian finite p-group G and the number of conjugacy class of it. 第二部分,我们建立了非交换有限p-群G与其共轭类个数之间的关系,并且得到了较小阶的p-群G与其共轭类个数之间一种更简洁的相互决定关系。
Finite Groups Whose Monolithic Characters Vanish on at Most Two Conjugacy Classes 每个Monolithic特征标至多零化两个共轭类的有限群
Let G be a finite group, | x | be the length of conjugacy clas containing x. 设G为一有限群,x为元素x所属共轭类的长度。
In particular, for a finite group G, every simple module of G is appeared in the conjugacy representation of G if and only if the sum of every row in the character table of G is not zero. 特别地,有限群的群代数的每一个单模均出现在共轭表示分解式中当且仅当有限群的特征标表中每一行元素之和不为0。