Derived norms of finite groups

作     者：Zhencai Shen Shirong Li Jiping Zhang 

作者机构：College of ScienceChina Agricultural UniversityBeijing100183China Department of MathematicsGuangxi UniversityNanning 530004China LMAM and School of Mathematical SciencesPeking UniversityBeijing 100871China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2022年第65卷第12期

页      面：2493-2502页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 11631001 and 12071181) 

主　　题：derived subgroups D-groups nilpotent residuals IO-D-groups 

摘      要：The intersection of particular subgroups is a kind of interesting substructure in group theory. Let G be a finite group and D(G) be the intersection of the normalizers of the derived subgroups of all the subgroups of G. A group G is called a D-group if G = D(G). In this paper, we determine the nilpotency class of the nilpotent residual G^(N) and investigate the structure of D(G) by a new concept called the IO-D-group. A non-D-group G is called an IO-D-group(inner-outer-D-group) if all of its proper subgroups and proper quotient groups are D-groups. The structure of IO-D-groups are described in detail in this paper. As an application of the classification of IO-D-groups, we prove that G is a D-group if and only if any subgroup of G generated by3 elements is a D-group.