A characterization of soluble groups in which normality is a transitive relation

A subgroup X ofý ýa group G is said to be an {\it{\scriptsize\calligra H}ý -ýsubgroup} ifý ýNG(X)∩Xg≤X for each element g belonging to Gý. ýIn [Mý. ýBianchi and e.a.ý, ýOn finite soluble groups in which normality is a transitive relationý, ý{\em Jý. ýGroup Theory}ý, ý{\bf 3} (2000) 147--156.] the authors showed that finite groups in which every subgroup has the {\scriptsize\calligra H}ý -ýproperty are exactly soluble groups in which normality is a transitive relationý. ýHere we extend this characterization to groups without simple sectionsý.