On finite \(T\)-groups and the Wielandt subgroup.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.20023
author: Gil Kaplan
publication date: 2012-05-07T00:00:00Z
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 6031519
zbMATH Keywords: Wielandt subgroup
zbMATH Keywords: Wielandt-type subgroup
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: T-groups
zbMATH Keywords: transitive normality
zbMATH Keywords: finite soluble groups
zbMATH Keywords: finite metanilpotent groups
zbMATH Keywords: weakly special elements
MaRDI profile type: Publication
full work available at URL: jgt.2011.082
OpenAlex ID: W2003421989
cites work: On finiteJ-groups
cites work: The intersection map of subgroups
cites work: On finite solvable groups in which normality is a transitive relation
cites work: Gruppen, in denen das Normalteilersein transitiv ist.
cites work: Finite Groups with Pro-Normal Subgroups
cites work: A Note on Finite Groups in Which Normality is Transitive
DOI: 10.1515/JGT.2011.082
Recommended article: On finiteJ-groups
Recommended article: On a class of locally finite T-groups
Recommended article: Tabor groups with finiteness conditions.
Recommended article: On the Wielandt subgroup of generalized FC-groups
Recommended article: Finite \(p\)-groups and \(k((t))\).
Recommended article: T-groups, polycyclic groups, and finite quotients.
Recommended article: On subgroups of finite \(p\)-groups.
Recommended article: Q3715315
Recommended article: Q3726330
Recommended article: On groups with finitely many derived subgroups.
title: On finite \(T\)-groups and the Wielandt subgroup.
published in: Journal of Group Theory
review text: All groups considered in the paper under review are finite. The `Wielandt subgroup' \(\omega(G)\) of a group \(G\) is the intersection of the normalisers of all subnormal subgroups of \(G\). Groups with \(\omega(G)=G\) are the groups in which every subnormal subgroup is normal, or, in other words, groups in which normality is transitive. These groups are known as `T-groups'.NEWLINENEWLINE In this paper, another Wielandt-type subgroup is defined. For a group \(G\), let \(\omega^*(G)\) be the set of all elements \(x\in G\) such that if \(x\in N_G(H)\), then \(x\in N_G(K)\) for each normal subgroup \(K\) of a subgroup \(H\) of \(G\). It is clear that \(\omega^*(G)\) is a subset of \(\omega(G)\). Moreover, \(\omega^*(G)\) is composed of all elements \(x\) of \(G\) such that \(x\in\omega(H)\) for each subgroup \(H\leq G\) such that \(x\in H\) (Lemma~2.1). Soluble T-groups are the groups such that \(\omega^*(G)=G\) (Lemma~2.2) and, for a metanilpotent group \(G\), \(\omega(G)=\omega^*(G)\). This allows the author to find some properties of the Wielandt subgroup for metanilpotent groups. For instance, if \(P\) is a Sylow \(p\)-subgroup of the metanilpotent group \(G\), \(p\) a prime, and \(P_0\leq P\), where \(P_0\) is a Sylow \(p\)-subgroup of \(\omega(G)\), then \(P_0\leq Z_2(P)\), and either \(\omega(G)\) is \(p\)-nilpotent, or \(P_0\) is a maximal Abelian subgroup of \(P\); furthermore, in the latter case \(P\) has class number at most~\(3\), and it is metabelian (Theorem~2).NEWLINENEWLINE Given a group \(G\), an element \(x\in G\) is said to be `special' when there exists a proper subgroup \(H\) of \(G\) such that \(H\) is normalised by \(x\) and \(x\) acts on \(H\) as a special automorphism (in the sense \(H^x=H\) and there exists a proper subgroup \(A\) of \(H\) such that \(H=AA^x\)). An element \(x\in G\) is `weakly special' when there exists a proper subgroup \(H\) of \(G\) such that \(H=AA^x\) for a certain \(A<H\). The set of special and weakly special elements …
reviewed by: R. Esteban-Romero
description: scientific article; zbMATH DE number 6031519