A new characterization of groups with central chief factors

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.