A new characterization of groups with central chief factors

Abstract

In [1] it is proved that a locally nilpotent group is an \((X)\)-group arising the question whether the converse holds. In this paper we derive some interesting properties and give a complete characterization of \((X)\)-groups. As a consequence we obtain a new characterization of groups whose chief factors are central and it follows also that there exists an \((X)\)-group which is not locally nilpotent, thus answering the question raised in [1]. We also prove a result  which extends one on finitely generated nilpotent groups due to Gruenberg.