Finite groups containing many subgroups are solvable
type: Item
instance of: scholarly article
title: Finite groups containing many subgroups are solvable
zbMATH Open document ID: 0854.20028
author: Marcel Herzog
author: P. Longobardi
author: M. Maj
author: Avinoam Mann
published in: Journal of Algebra
publication date: 1996-06-25T00:00:00Z
review text: The aim of this paper is to obtain sufficient conditions for the solubility of a finite group \(G\) in terms of its Hall \(\pi\)-structure. Here are three samples selected from a wealth of interesting results: Theorem 6. If \(m\) is an integer, \(2\leq m\leq\pi(G)-1\) and if \(G\) contains a Hall \(\{p_1,p_2,\dots,p_m\}\)-subgroup for all \(m\)-tuples of distinct primes in \(\pi(G)\), then \(G\) is soluble. Corollary 7. If \(G\) contains Hall \(\{p,q\}\)-subgroups for all pairs of primes dividing the order of \(G\), with the possible exception of \(\{2,3\}\) and \(\{2,5\}\) then \(G\) is soluble. Theorems 11, 12. Let \(G\) be a finite group, let \(n=\pi(G)\) and let \(k\) be an integer, \(k<\max(n,2^{n-4}+2)\). Suppose that \(G\) contains subgroups of order \(d\) for all divisors \(d\) of \(|G|\) with the exception of at most \(k\) such divisors. Then \(G\) is soluble.
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 868026
zbMATH Keywords: soluble groups
zbMATH Keywords: solubility of finite groups
zbMATH Keywords: Hall \(\pi\)-structure
zbMATH Keywords: Hall \(\{p,q\}\)-subgroups
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
full work available at URL: jabr.1996.0042
OpenAlex ID: W2082742182
DOI: 10.1006/JABR.1996.0042
description: scientific article; zbMATH DE number 868026