A subgroup H of a group G is nearly normal if it has finite index in its normal closure . 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-1-2,
author = {Francesco de Giovanni and Carmela Musella},
title = {Groups with nearly modular subgroup lattice},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {13-20},
zbl = {0985.20016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-1-2}
}
Francesco de Giovanni; Carmela Musella. Groups with nearly modular subgroup lattice. Colloquium Mathematicae, Tome 89 (2001) pp. 13-20. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-1-2/