On structure analysis of finitely generated abelian groups

In this paper we give an overview of the structural analysis of finitely generated abelian groups and modules. We give an overview of recent results on the structure theory of these objects in various situations, in particular in the case of torsionfree groups of infinite rank. We also mention several open problems. We also propose a method to study the structure of finitely generated abelian groups and finitely generated nilpotent groups by using some invariants of their derived series. This method provides a new framework for studying the structure of finitely generated abelian groups and finitely generated nilpotent groups.