On Local Nilpotency of the Normal Subgroups of a Group  

作　　者：Zhirang Zhang Jiachao Li 

机构地区：[1]School of Applied Mathematics, Chengdu University of Information Technology Chengdu 610225, China

出　　处：《Algebra Colloquium》2016年第3期531-540,共10页代数集刊（英文版）

基　　金：Project supported by the National Natural Science Foundation of China （Nos. 11371335, 11471055）.

摘　　要：A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a group G is said to have property v if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP（G） is the Hirsch-Plotkin radical of G, and φf （G） is the intersection of all the maximal subgroups of finite index in G （here φf（G） = G if no such maximal subgroups exist）. It is shown that a group G has property μ if and only if HP（G/φf（G）） = HP（G）/φf（G） and that the class of groups with property v is a proper subclass of that of groups with property it. Also, the structure of the normal subgroups of a group： nilpotency with the minimal condition, is investigated.A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a group G is said to have property v if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP（G） is the Hirsch-Plotkin radical of G, and φf （G） is the intersection of all the maximal subgroups of finite index in G （here φf（G） = G if no such maximal subgroups exist）. It is shown that a group G has property μ if and only if HP（G/φf（G）） = HP（G）/φf（G） and that the class of groups with property v is a proper subclass of that of groups with property it. Also, the structure of the normal subgroups of a group： nilpotency with the minimal condition, is investigated.

关 键 词：normal subgroups nilpotent groups locally nilpotent groups Frattini sub-group Hirsch-Plotkin radical 

分 类 号：O157.2[理学—数学] O152.1[理学—基础数学]