Finite groups and subgroup-permutability
type: Item
instance of: scholarly article
title: Finite groups and subgroup-permutability
zbMATH Open document ID: 0849.20014
DOI: 10.1007/BF01759356
author: Mariagrazia Bianchi
author: Anna Gillio Berta Mauri
author: Libero Verardi
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1996-07-08T00:00:00Z
review text: For the purposes of this paper, a finite group \(G\) is defined to be irreducible if the permutability graph on non-normal subgroups is connected, and reducible otherwise. The maximum diameter of the components of this graph is denoted \(\Delta(G)\). The four main results of the paper are: A: If \(G\) is soluble, then \(\Delta(G)\leq 4\). B: If \(G\) is non-abelian simple, then it is irreducible and \(\Delta(G)\leq 16\). C: If \(G\) is reducible, then it is soluble. D: If \(G\) is non-soluble, then \(\Delta(G)\leq\max\{\Delta(S)\}+4\), as \(S\) ranges over all non-abelian simple groups. The proof of B, C and D uses the classification theorem for finite simple groups.
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 878432
zbMATH Keywords: finite groups
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: soluble groups
zbMATH Keywords: permutability graph on non-normal subgroups
zbMATH Keywords: simple groups
reviewed by: Robert A. Wilson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 878432