Some notes on the minimal subgroups of Fitting subgroups of finite groups
type: Item
instance of: scholarly article
title: Some notes on the minimal subgroups of Fitting subgroups of finite groups
zbMATH Open document ID: 1008.20011
DOI: 10.1016/S0022-4049(01)00126-8
published in: Journal of Pure and Applied Algebra
publication date: 2002-08-22T00:00:00Z
review text: Many authors have studied the structure of a finite group \(G\), under the assumption that all minimal subgroups of \(G\) are well-situated in the group. First results in this line are due to Itô and Buckley. Recently, \textit{Y. Wang} and the reviewer [J. Pure Appl. Algebra 153, No. 2, 121-127 (2000; Zbl 0967.20009)] studied the influence of c-normality of the minimal subgroups on the structure of \(G\) through the theory of formations and obtained many new results which generalize some classical and previous ones. The present paper represents an attempt to extend and improve the results of the above mentioned authors by studying the influence of the minimal subgroups of the Fitting subgroup of \(G\). They consider saturated formations \(\mathcal F\) containing the class of nilpotent groups and look for conditions that guarantee the group \(G\) is in \(\mathcal F\) if a fixed quotient \(G/H\), where \(H\) is a normal subgroup of \(G\) enjoying certain embedding properties concerning c-normality and \(\mathcal F\)-hypercentrality, belongs to the class \(\mathcal F\). For instance Theorem 1 states that, in this situation, the following are equivalent: (i) G is in \(\mathcal F\) (ii) Every element of order 4 of \(F(H)\) is c-normal in \(G\) and \(\langle x\rangle\) lies in the \(\mathcal F\)-hypercenter of \(G\) for every element \(x\) of prime order of \(F(H)\).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 1787454
zbMATH Keywords: finite groups
zbMATH Keywords: minimal subgroups
zbMATH Keywords: saturated formations
zbMATH Keywords: Fitting subgroups
zbMATH Keywords: hypercenter
author: Yangming Li
reviewed by: Adolfo Ballester-Bolinches
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787454