Finite groups and $k$-submodular subgroups

TL;DR

This paper introduces and explores the concepts of $k$-submodular subgroups and related group classes, generalizing existing subgroup notions to analyze their properties and classifications within finite groups.

Contribution

It defines new subgroup concepts like $k$-submodular subgroups and $k$-$ ext{LM}$-groups, extending classical subgroup theories and investigating their properties in finite groups.

Findings

Characterization of groups with systems of $k$-submodular subgroups

Generalization of modular and submodular subgroup concepts

Analysis of classes of groups based on $k$-submodular subgroup systems

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.