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

摘要

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 N whose proper subgroups of cardinality N 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 N whose proper normal subgroups of cardinality N have the T-property, then every subnormal subgroup of G has only finitely many conjugates.