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

A subgroup X of‎ ‎a group G is said to be an H-‎subgroup if‎ ‎NG(X) ∩ Xg  ≤ X for each element g belonging to G‎. ‎In [M‎. ‎Bianchi and e.a.‎, ‎On finite soluble groups in which normality is a transitive relation‎, J‎. ‎Group Theory, ‎ ۳ (۲۰۰۰) ۱۴۷--۱۵۶.] the authors showed that finite groups in which every subgroup has the H‎-‎property are exactly soluble groups in which normality is a transitive relation‎. ‎Here we extend this characterization to groups without simple sections‎.

Keywords:

‎H-subgroups , T‎-groups , ‎ ‎pronormal subgroups‎ , ‎weakly normal subgroups‎ , ‎pronorm and H-‎norm of a group