引用本文:廖俊梅, 曹洪平.23p阶群的共轭类个数与群的结构[J].西南大学学报(自然科学版),2018,40(12):86~89
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23p阶群的共轭类个数与群的结构
廖俊梅, 曹洪平
西南大学 数学与统计学院, 重庆 400715
摘要:
在有限群的研究中,群的阶和群的元素的共轭类个数是群的两个非常重要的数量,这两个数量对群的结构性质有很大影响,很多有限群完全可由这两个数量确定.对于阶为23pp为奇素数)的有限群,根据分类定理,一共有19类互不同构的23p阶群,对这19类23p阶群,利用它们的生成元与生成关系及数论和群论知识确定了它们的共轭类,并由此得出了它们的共轭类的个数,进一步对这19类23p阶群的共轭类个数进行比较,得到了23p阶群的共轭类个数的最小值.反过来,根据已得到的23p阶群的共轭类个数的最小值及群的阶,利用23p阶群的分类,对19类23p阶群的共轭类个数进行比较,确定了共轭类个数取最小值的23p阶群的具体结构.
关键词:  共轭类个数  最小值  群结构
DOI:10.13718/j.cnki.xdzk.2018.12.013
分类号:O152.1
基金项目:国家自然科学基金项目(11471266,11271301).
The Number of Conjugate Classes and the Structure of the Group with Order 23p
LIAO Jun-mei, CAO Hong-ping
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Abstract:
In the study of finite groups, the order of a group and the number of conjugate classes of the elements of a group are two very important quantities of the group. These two quantities have a great influence on the structure and properties of the group. Many finite groups can be determined completely by these two quantities. For groups of order 23p (p is an odd prime), the number of conjugate classes of 19 kinds of groups of order 23p is determined by using their generators and generative relations as well as the knowledge of number theory and group theory. The number of conjugate classes of the 19 kinds of groups of order 23p is further compared, and the minimum number of conjugate classes of groups of order 23p is obtained. According to the minimum number of conjugate classes and the order of groups of order 23p, the number of conjugate classes of 19 groups of order 23p is compared by using the classification of groups of order 23p, and the concrete structure of groups of order 23p with the minimum number of conjugate classes is determined.
Key words:  conjugate class number  minimum value  group structure
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