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Finite groups and $k$-submodular subgroups

T. I. Vasilyeva

TL;DR

The paper develops a framework for $k$-submodular subgroups in finite groups by introducing $n$-modularly embedded subgroups and $k$-LM-groups as generalizations of classical notions. It formulates a formation-theoretic approach, defines four pivotal classes $\mathfrak{X},\mathfrak{Y},\mathfrak{K},\mathfrak{F}$, and establishes their hierarchical inclusions and equalities $\mathfrak{K}=\mathfrak{U}\cap \mathrm{w}\mathfrak{U}_{k}$ and $\mathfrak{F}=\mathrm{w}\mathfrak{K}$. The main results characterize when all maximal subgroups are $k$-submodular in terms of supersolubility and constrained chief-factor actions, and prove that $LF(h)=\mathfrak{K}$ and related local formations coincide with these classes. These findings extend the theory of submodular/LM-groups and provide a structured classification for finite groups under $k$-submodular subgroup conditions, with implications for the subnormality of Sylow subgroups and the composition structure of groups in these formations.

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.

Finite groups and $k$-submodular subgroups

TL;DR

The paper develops a framework for -submodular subgroups in finite groups by introducing -modularly embedded subgroups and -LM-groups as generalizations of classical notions. It formulates a formation-theoretic approach, defines four pivotal classes , and establishes their hierarchical inclusions and equalities and . The main results characterize when all maximal subgroups are -submodular in terms of supersolubility and constrained chief-factor actions, and prove that and related local formations coincide with these classes. These findings extend the theory of submodular/LM-groups and provide a structured classification for finite groups under -submodular subgroup conditions, with implications for the subnormality of Sylow subgroups and the composition structure of groups in these formations.

Abstract

For natural numbers and , the concepts of -modularly embedded subgroup, -submodular subgroup and --group are given, which generalize, respectively, the concepts of modular subgroup, submodular subgroup and -group. Classes of groups with given systems of -submodular subgroups are investigated.
Paper Structure (4 sections, 29 theorems, 9 equations)

This paper contains 4 sections, 29 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of $k$-submodular subgroups of groups
  4. Proofs of main results

Key Result

Theorem 1

Let $G$ be a group. Then the following statements are equivalent. $(1)$ Every maximal subgroup of $G$ is $k$-submodular in $G$. $(2)$$G$ is supersoluble and for all complemented chief factors $H/K$ of $G$, $G/C_{G}(H/K)$ is either $1$ or cyclic of order $q^{n}$ where $q$ is some prime and $n$ is som

Theorems & Definitions (54)

  • Definition 1
  • Definition 2
  • Example 1
  • Remark 1
  • Theorem 1
  • Corollary 1
  • Definition 3
  • Example 2
  • Remark 2
  • Theorem 2
  • ...and 44 more