On Local Nilpotency of the Normal Subgroups of a Group

Abstract

A group G is said to have property μ whenever N is a non-locally nilpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property ν satisfy property μ, where a group G is said to have property ν 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 ν is a proper subclass of that of groups with property μ. Also, the structure of the normal subgroups of a group: nilpotency with the minimal condition, is investigated.