Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On generalizations of Iwasawa's theorem

Jiangtao Shi, Fanjie Xu, Mengjiao Shan

Abstract

Iwasawa's theorem indicates that a finite group $G$ is supersolvable if and only if all maximal chains of the identity in $G$ have the same length. As generalizations of Iwasawa's theorem, we provide some characterizations of the structure of a finite group $G$ in which all maximal chains of every minimal subgroup have the same length. Moreover, let $δ(G)$ be the number of subgroups of $G$ all of whose maximal chains in $G$ do not have the same length, we prove that $G$ is a non-solvable group with $δ(G)\leq 16$ if and only if $G\cong A_5$.

On generalizations of Iwasawa's theorem

Abstract

Iwasawa's theorem indicates that a finite group is supersolvable if and only if all maximal chains of the identity in have the same length. As generalizations of Iwasawa's theorem, we provide some characterizations of the structure of a finite group in which all maximal chains of every minimal subgroup have the same length. Moreover, let be the number of subgroups of all of whose maximal chains in do not have the same length, we prove that is a non-solvable group with if and only if .
Paper Structure (6 sections, 13 theorems)

This paper contains 6 sections, 13 theorems.

Table of Contents

  1. Introduction
  2. Some necessary lemmas
  3. Proof of Theorem
  4. Proof of Theorem
  5. Proof of Theorem
  6. Remarks

Key Result

Proposition 1.1

Let $G$ be a group, $H<K<G$. If all maximal chains of $H$ in $G$ have the same length, then $(1)$ all maximal chains of $H$ in $K$ have the same length; $(2)$ all maximal chains of $K$ in $G$ have the same length.

Theorems & Definitions (29)

  • Proposition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 19 more