Finite groups and $k$-submodular subgroups

Abstract

For natural numbers $n$ and $k$, the concepts of $n$-modularly embedded subgroup, $k$-submodular subgroup and $k$-$\mathrm{LM}$-group are given, which generalize, respectively, the concepts of modular subgroup, submodular subgroup and $\mathrm{LM}$-group. Classes of groups with given systems of $k$-submodular subgroups are investigated.

16 pages. This article is an expanded and revised version. Definitions 1 and 2 have been clarified to remove unnecessary conditions. The main results are supplemented either by new equivalent conditions or by new items. Examples and comments are given. A new concept of k-LM-group has been introduced. Some new results have been proven. New sources have been added to References