共轭类的长和有限群的结构

The Length of Conjugacy Classes and the Structure of Finite Groups

设Ｇ是有限群，并设Ｃｏｎ（Ｇ）是由Ｇ的一切共轭类组成的集合。本文目的是考察Ｃｏｎ（Ｇ）的算术结构对Ｇ的群结构的影响。我们着重于Ｃｏｎ（Ｇ）的ｐ－部分结构，并得到关于ｐ－幂零群的两个定理。这里，ｐ表示一个有限群的阶的最小素因子，用这两个定理，我们还得到若干结果，其中两个改进了Ｄ．Ｃｈｉｌｌａｇ和Ｍ．Ｈｅｒｚｏｇ关于共轭类长的两个结果．

Let G be a finite group, and let Con(G) be the set of all the con-jugacy classes of G. The purpose of this paper is to investigate the influence of the arithmetical structure of Con(G) on the group structure of G. We put our emphasis on the p-part structure of Con(G),and obtain two theorem:on p-nilpotent groups.Here,p denotes the least prime divisor of order of a finite group.Using the two theorems, we also get several results,two of which improve the two results of D. Chillag and M. Herzog on the length of conjugacy classes.