Groups with nearly modular subgroup lattice
A subgroup $H$ of a group $G$ is nearly normal if it has finite index in its normal closure $H^G$. A relevant theorem of B. H. Neumann states that groups in which every subgroup is nearly normal are precisely those with finite commutator subgroup. We shall say that a subgroup $H$ of a group $G$ is nearly modular if $H$ has finite index in a modular element of the lattice of subgroups of $G$. Thus nearly modular subgroups are the natural lattice-theoretic translation of nearly normal subgroups. In this article we study the structure of groups in which all subgroups are nearly modular, proving in particular that a locally graded group with this property has locally finite commutator subgroup.