Groups in which the hypercentral factor group is a T-group.
type: Item
instance of: scholarly article
title: Groups in which the hypercentral factor group is a T-group.
zbMATH Open document ID: 1119.20023
author: J. C. Beidleman
author: Hermann Heineken
published in: Ricerche di Matematica
publication date: 2007-07-19T00:00:00Z
review text: The authors consider finite \(\text{T}_1\)-groups, i.e., the finite groups in which the factor group by its hypercenter is a group with transitivity of normality (T-groups). They describe some basic properties of such groups and highlight connections with its Frattini subgroup and its nilpotent radical. Here are some of the main results: Theorem A. Let \(G\) be a soluble finite \(\text{T}_1\)-group and \(R\) its nilpotent residual. Then 1. \(R\) is nilpotent of class at most 2. 2. \(G\) acts by conjugation on the factor-group of \(R\) by its derived subgroup as a group of power automorphisms. 3. The nilpotent residual of \(G^2\) is Abelian. Theorem D. Let \(G\) be a finite soluble group that is a product of two mutually permutable subgroups \(A\) and \(B\). If \(A\) is soluble and \(B\) is a \(\text{T}_1\)-group such that its nilpotent residual is \(G\)-invariant, then \(G\) is supersoluble. Corollary. The class of solvable \(\text{T}_1\)-groups is contained in the Fitting core of the formation of supersoluble groups.
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5173067
zbMATH Keywords: hypercenters
zbMATH Keywords: T-groups
zbMATH Keywords: Fitting cores
zbMATH Keywords: Frattini subgroup
zbMATH Keywords: nilpotent radicals
zbMATH Keywords: mutually permutable products
zbMATH Keywords: supersolvability
zbMATH Keywords: finite supersolvable groups
zbMATH Keywords: formations
zbMATH Keywords: transitive normality
reviewed by: Igor Subbotin
MaRDI profile type: Publication
full work available at URL: s11587-006-0012-z
OpenAlex ID: W2052274328
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cites work: ON PRODUCTS OF GROUPS FOR WHICH NORMALITY IS A TRANSITIVE RELATION ON THEIR FRATTINI FACTOR GROUPS
DOI: 10.1007/S11587-006-0012-Z
description: scientific article; zbMATH DE number 5173067