On the existence of solvable normal subgroups in finite groups
type: Item
instance of: scholarly article
title: On the existence of solvable normal subgroups in finite groups
zbMATH Open document ID: 0929.20018
DOI: 10.1007/BF02355085
published in: Mathematical Notes
publication date: 1999-03-23T00:00:00Z
review text: Let \(G\) be a finite group and \(p\) a prime dividing the order of \(G\). If (i) \(G\) is \(p\)-soluble, (ii) \(x\) is a \(p\)-element of \(G\), and (iii) \(\langle x,x^g\rangle\) is a group of odd order for every \(g\in G\), then \(x\in O(G)\) (Theorem 1). If (i) all Sylow \(p\)-subgroups of \(G\) are Abelian, (ii) \(x,y\) are non-identity elements of \(G\), (iii) \(\langle x^g,y^h\rangle\) is a \(p\)-group for all elements \(g,h\in G\), and (iv) \(\langle x^G\rangle=\langle y^G\rangle\), then \(\langle x,x^g\rangle\subseteq O_p(G)\) (Theorem 2).
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 1238052
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: \(p\)-elements
zbMATH Keywords: finite groups
author: V. N. Tyutyanov
MaRDI profile type: Publication
cites work: Q3941595
cites work: Q4006528
cites work: Simple Groups and Simple Lie Algebras
cites work: The local structure of finite groups of characteristic 2 type
description: scientific article; zbMATH DE number 1238052