Groups with nearly modular subgroup lattice

Francesco de Giovanni¹ ; Carmela Musella¹

¹ Dipartimento di Matematica e Applicazioni Università di Napoli “Federico II” Complesso Universitario Monte S. Angelo Via Cintia I-80126 Napoli, Italy

Colloquium Mathematicum, Tome 88 (2001) no. 1, pp. 13-20

Résumé

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.

Détail

DOI : 10.4064/cm88-1-2

Keywords: subgroup group nearly normal has finite index its normal closure relevant theorem neumann states groups which every subgroup nearly normal precisely those finite commutator subgroup shall say subgroup group nearly modular has finite index modular element lattice subgroups nearly modular subgroups natural lattice theoretic translation nearly normal subgroups article study structure groups which subgroups nearly modular proving particular locally graded group property has locally finite commutator subgroup