Finite groups in which \(\tau\)-quasinormality is a transitive relation.
type: Item
instance of: scholarly article
title: Finite groups in which \(\tau\)-quasinormality is a transitive relation.
zbMATH Open document ID: 1217.20012
DOI: 10.4171/RSMUP/124-16
author: V. O. Lukyanenko
author: A. N. Skiba
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2011-02-10T00:00:00Z
full work available at URL: https://eudml.org/doc/239759
full work available at URL: http://rendiconti.math.unipd.it/volumes/vol124.php
review text: In the paper and in this review we deal only with finite groups. The well-known fact that normality is not a transitive relation in general has motivated the study of groups in which normality is a transitive relation, the T-groups. Later the condition of normality has been replaced by other weaker properties, like permutability or Sylow permutability (S-permutability in short). Recall that a subgroup \(H\) of a group \(G\) is permutable (respectively, S-permutable or S-quasinormal) in \(G\) when \(H\) permutes with all subgroups (respectively, all Sylow subgroups) of \(G\). Groups in which permutability (respectively, S-permutability) is transitive have been called PT-groups (respectively, PST-groups). The classes of T-groups, PT-groups, and PST-groups have been widely studied. The present paper deals with the effect of imposing on a subgroup to permute with the members of a smaller family of subgroups. For instance, a subgroup \(H\) of a group \(G\) is S-semipermutable (or S-semiquasinormal or S-seminormal) in \(G\) when \(H\) permutes with all Sylow \(p\)-subgroups \(P\) of \(G\) for the primes \(p\) not dividing \(|H|\). The authors introduced [in Acta Math. Hung. 125, No. 3, 237-248 (2009; Zbl 1207.20009)] another interesting embedding property: a subgroup \(H\) of a group \(G\) is \(\tau\)-quasinormal in \(G\) when \(H\) permutes with all Sylow \(p\)-subgroups \(P\) of \(G\) such that \(p\) does not divide \(|H|\) but \((|H|,|P^G|)\neq 1\). S-permutability implies S-semipermutability and S-semipermutability implies \(\tau\)-quasinormality, but none of the converses is true. As a generalisation of T-groups, in this paper groups in which \(\tau\)-quasinormality is a transitive relation are considered. They are called TQT-groups and characterised as the supersoluble groups \(G\) which have an Abelian normal Hall subgroup of odd order \(D\) such that \(G/D\) is nilpotent, every subgroup of \(D\) is normal in \(G\), and every subgroup of \(G\) is \(\tau\)-quasinorm…
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D35
zbMATH DE Number: 5849507
zbMATH Keywords: finite groups
zbMATH Keywords: quasinormal subgroups
zbMATH Keywords: TQT-groups
zbMATH Keywords: PST-groups
zbMATH Keywords: transitive quasinormality
zbMATH Keywords: Sylow permutability
zbMATH Keywords: semipermutable subgroups
zbMATH Keywords: supersoluble groups
zbMATH Keywords: Hall subgroups
reviewed by: R. Esteban-Romero
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5849507