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

Abstract

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, ‎ 3 (2000) 147--156.] 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