Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Indices of non-supersolvable maximal subgroups in finite groups

Antonio Beltrán, Changguo Shao

TL;DR

The paper classifies non-solvable finite groups whose maximal subgroups are either supersolvable or have index equal to a prime or the square of a prime. Using foundational results from Doerk–Hall, Guralnick, Demina–Maslova, and related wreath-product analyses, it shows that the solvable radical $S(G)$ is supersolvable and that $G/S(G)$ falls into a short catalog of structures: either a small set of non-abelian simple or almost-simple groups, or a 2-extension of a direct product of ${ m PSL}_2(p^{2^a})$-type factors. The main theorem enumerates precisely the possibilities for $G/S(G)$ and for the $2$-extensions, with a corollary identifying the four non-abelian simple groups that can occur when maximal subgroups of prime or squared prime index are non-supersolvable. The results rely on classic classifications and a detailed analysis of normal subgroups and wreath-product behavior to constrain the group structure under the stated maximal-subgroup hypotheses.

Abstract

Two classic results, due to K. Doerk and P. Hall respectively, establish the solvability of those finite groups all of whose maximal subgroups are supersolvable, and the solvability of finite groups in which all maximal subgroups have prime or squared prime index. In this note we describe the structure of the non-solvable finite groups whose maximal subgroups are either supersolvable or have prime or squared prime index.

Indices of non-supersolvable maximal subgroups in finite groups

TL;DR

The paper classifies non-solvable finite groups whose maximal subgroups are either supersolvable or have index equal to a prime or the square of a prime. Using foundational results from Doerk–Hall, Guralnick, Demina–Maslova, and related wreath-product analyses, it shows that the solvable radical is supersolvable and that falls into a short catalog of structures: either a small set of non-abelian simple or almost-simple groups, or a 2-extension of a direct product of -type factors. The main theorem enumerates precisely the possibilities for and for the -extensions, with a corollary identifying the four non-abelian simple groups that can occur when maximal subgroups of prime or squared prime index are non-supersolvable. The results rely on classic classifications and a detailed analysis of normal subgroups and wreath-product behavior to constrain the group structure under the stated maximal-subgroup hypotheses.

Abstract

Two classic results, due to K. Doerk and P. Hall respectively, establish the solvability of those finite groups all of whose maximal subgroups are supersolvable, and the solvability of finite groups in which all maximal subgroups have prime or squared prime index. In this note we describe the structure of the non-solvable finite groups whose maximal subgroups are either supersolvable or have prime or squared prime index.
Paper Structure (3 sections, 7 theorems, 4 equations)

This paper contains 3 sections, 7 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof

Key Result

Theorem 2.1

Gura Let $G$ be a non-abelian simple group with $H < G$ and $|G : H| = p^a$, $p$ prime. One of the following holds.

Theorems & Definitions (11)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Theorem 2.6
  • Lemma 2.7
  • Remark 2.8
  • proof : Proof of Theorem A
  • ...and 1 more