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

Abstract

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.