Uncountable groups in which normality is a transitive relation

Abstract

A group is called a [Formula: see text]-group if all its subnormal subgroups are normal. It is proved here that if [Formula: see text] is an uncountable soluble periodic group of regular cardinality [Formula: see text] in which all subnormal subgroups of cardinality [Formula: see text] are normal, then [Formula: see text] is a [Formula: see text]-group, provided that it satisfies a suitable residual condition. An example shows that this latter restriction cannot be avoided.