Characterization of \(r\)-solvable groups
type: Item
instance of: scholarly article
title: Characterization of \(r\)-solvable groups
zbMATH Open document ID: 0956.20006
published in: Siberian Mathematical Journal
publication date: 2000-11-21T00:00:00Z
full work available at URL: https://eudml.org/doc/120814
review text: The following theorem is proven which generalizes a result by \textit{G.~Glauberman} [Ill. J. Math. 12, 76-98 (1968; Zbl 0182.35502)]: Let \(G\) be a finite \(K\)-group and let \(r\) be a prime divisor of the order of \(G\). Then \(G\) is \(r\)-soluble if and only if every pair of elements in \(G\) generates an \(r\)-soluble subgroup. Recall that a group \(G\) is called \(K\)-group if all its composition factors are known simple groups. First, the author proves that a minimal counterexample must be a finite simple group. Then the author proves that every known simple group is either \(r\)-soluble or contains two \(r\)-elements which generate a non-\(r\)-soluble group.
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 1534387
zbMATH Keywords: prime graphs
zbMATH Keywords: Gruenberg-Kegel graphs
zbMATH Keywords: \(p\)-soluble groups
zbMATH Keywords: finite simple groups
zbMATH Keywords: finite groups of Lie type
author: V. N. Tyutyanov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1534387