On finite soluble groups which behave like nilpotent groups with respect to the Frattini group
type: Item
instance of: scholarly article
title: On finite soluble groups which behave like nilpotent groups with respect to the Frattini group
zbMATH Open document ID: 0806.20021
published in: Journal of Algebra
publication date: 1994-08-16T00:00:00Z
review text: It is known that if \(G\) is a finite nilpotent group with subgroups \(H\) and \(K\), \(K\) normal in \(G\), then \(\Phi(H) \leq \Phi(G)\) and \(\Phi(G/K) = \Phi(G) \cdot K/K\). In this interesting note the author characterizes finite solvable groups which possess these \(\Phi\)-features. The nature of the results is indicated by Theorem \(1'\) (stated in the vernacular by the author). Let \(G\) be a finite solvable group. The following statements are equivalent: (a) If \(H\) and \(K\) are subgroups of \(G\) with \(H \leq K\) then \(\Phi(H) \leq \Phi(K)\). (b) \(G/\Phi(G)\) has elementary abelian Sylow subgroups. (c) If \(H/K\) is a \(p\)-chief factor of \(G\) then \(G/C_ G(H/K)\) is a \(p'\)-group with elementary abelian Sylow subgroups. (d) If \(H \leq G\) then \(\Phi(H) \leq \Phi(G)\). (e) If \(P\) is a Sylow subgroup of \(G\) then \(\Phi(P) \leq \Phi(G)\). In general the terminology and methods of the theory of classes of groups are used.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 614222
zbMATH Keywords: finite nilpotent group
zbMATH Keywords: finite solvable groups
zbMATH Keywords: elementary abelian Sylow subgroups
zbMATH Keywords: \(p\)-chief factor
author: Klaus Doerk
reviewed by: W. E. Deskins
MaRDI profile type: Publication
full work available at URL: jabr.1994.1199
OpenAlex ID: W2054432561
DOI: 10.1006/JABR.1994.1199
description: scientific article; zbMATH DE number 614222