共轭长度、特征标次数与有限群的可解性

Conjugation lengths,character degrees and solvability of finite groups

对一个正整数n,若所有共轭数长度都与其互素的有限群均可解.则称n为共轭类可解互素数,简记为CSC-数;类似地,将此定义中的"共轭数长度"替换为"不可约复特征标次数",则称n为特征标次数可解互素数,简记为DSC-数.同时证明了,正整数n是CSC-数(DSC-数)的充要条件是n能被2或15整除.

The positive integer n is called conjugation class of solvable coprime number（abbreviated CSC-number） if all the conjugation class lengths and their coprime finite group are solvable;similarly,if ＂conjugation class lengths＂ of the definition is replaced by ＂irreducible complex character degrees＂,then n is called the solvable coprime number of irreducible complex character degrees（abbreviated DSC-number）.In this paper,it is proved that n being divided exactly by 2 or 15 is the necessary and sufficient condition for n being a CSC-number（DSC-number）.