Lattice characterization of some classes of groups by series of subgroups

Abstract

In this paper, we characterize several classes of groups by the properties of their weak congruence lattices. Namely, we give necessary and sufficient conditions for the weak congruence lattice of a group, under which this group is a [Formula: see text]-group, [Formula: see text]-group, metacyclic, cocyclic, hyperabelian, polycyclic, [Formula: see text]-group and [Formula: see text]-group. We also discuss groups for which all subgroups are simple, like the Tarski monsters, and we show that they represent a class of non-Dedekind groups with a particular embedding property for lattices of normal subgroups. All the mentioned characterized groups are related to (different) series of subgroups, and we represent these series as chains of intervals in the weak congruence lattice. The corresponding conditions are formulated in a purely lattice-theoretic language.