On some new developments in the theory of subgroup lattices of groups

A rather natural way for trying to obtain a lattice-theoretic characterization of a class of groups {\mathcal X} is to replace the concepts appearing in the definition of {\mathcal X} by lattice-theoretic concepts. The first to use this idea were Kontorovi\v{c} and Plotkin who in ۱۹۵۴ introduced the notion of modular chain in a lattice, as translation of a central series of a group, to determine a lattice-theoretic characterization of the class of torsion-free nilpotent groups. The aim of this paper is to present a recent application of this translation method to some generalized nilpotency properties.

Keywords:

modular lattice , permodular subgroup , permodular chain