Abstract
The aim of this paper is to investigate the behaviour of projective images of the groups which are finite over a term of their upper central series. In particular, we prove that for any positive integer k, the class of finitely generated groups in which the k-th term of the upper central series has finite index can be described in terms of lattice invariants, and so it is invariant under projectivities. In this context, we also study groups that have only finitely many maximal subgroups which are not permodular.
Highlights
One of the main problems in the study of subgroup lattices of groups is the translation of concepts and results of abstract group theory by means of purely lattice-theoretic objects, a crucial role in this context being played by modularity
The subgroup lattice of any abelian group is obviously modular and it is known that finite groups with a modular subgroup lattice are soluble, but there exist infinite simple groups with modular subgroup lattice, like for instance Tarski groups, i.e., infinite simple groups all of whose proper non-trivial subgroups
Let G be a group and let X be a permodular subgroup of G
Summary
One of the main problems in the study of subgroup lattices of groups is the translation of concepts and results of abstract group theory by means of purely lattice-theoretic objects, a crucial role in this context being played by modularity. Let G be a group containing a k-permodularly embedded subgroup of finite index for some positive integer k. Let G be a group and let X be a permodular subgroup of G such that the interval [G/X] is an infinite distributive lattice satisfying the maximal condition.
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