A NOTE ON GROUPS WHOSE PROPER LARGE SUBGROUPS HAVE A TRANSITIVE NORMALITY RELATION

A group $G$ is said to have the $T$ -property (or to be a $T$ -group) if all its subnormal subgroups are normal, that is, if normality in $G$ is a transitive relation. The aim of this paper is to investigate the behaviour of uncountable groups of cardinality  $\aleph$ whose proper subgroups of cardi...

Full description

Saved in:
Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society 2017-02, Vol.95 (1), p.38-47
Main Authors: DE GIOVANNI, FRANCESCO, TROMBETTI, MARCO
Format: Article
Language:English
Subjects:
Subgroups
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A group $G$ is said to have the $T$ -property (or to be a $T$ -group) if all its subnormal subgroups are normal, that is, if normality in $G$ is a transitive relation. The aim of this paper is to investigate the behaviour of uncountable groups of cardinality  $\aleph$ whose proper subgroups of cardinality $\aleph$ have a transitive normality relation. It is proved that such a group  $G$ is a $T$ -group (and all its subgroups have the same property) provided that $G$ has an ascending subnormal series with abelian factors. Moreover, it is shown that if $G$ is an uncountable soluble group of cardinality $\aleph$ whose proper normal subgroups of cardinality  $\aleph$ have the $T$ -property, then every subnormal subgroup of $G$ has only finitely many conjugates.
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972716000848