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Solvability of a group based on its number of subgroups

Angsuman Das, Arnab Mandal

Abstract

In this paper, we provide some conditions of (super)-solvability and nilpotency of a finite group $G$ based on its number of subgroups $Sub(G)$. Our results generalize the classification of finite groups with less than $20$ subgroups by Betz and Nash. We also provide an application of our results in studying comaximal subgroup graph of a group. Finally, we conclude with some open issues.

Solvability of a group based on its number of subgroups

Abstract

In this paper, we provide some conditions of (super)-solvability and nilpotency of a finite group based on its number of subgroups . Our results generalize the classification of finite groups with less than subgroups by Betz and Nash. We also provide an application of our results in studying comaximal subgroup graph of a group. Finally, we conclude with some open issues.
Paper Structure (8 sections, 20 equations, 1 figure, 1 table)

This paper contains 8 sections, 20 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Motivation
  3. Our Contribution
  4. Solvable Enforcing Numbers
  5. Supersolvable Enforcing Numbers
  6. Nilpotent Enforcing Numbers
  7. An Application to Comaximal Subgroup Graph of a Group
  8. Conclusion and Open Issues

Figures (1)

  • Figure 1: Enforcing Numbers $\leq 76$