Groups in which normality is a weakly transitive relation

We study groups in which normality is a weakly transitive relation,giving an extension of Theorem A [On finiteT-groups, J. Aust. Math.Soc.75(2003)181–191] due to 17 Ballester-Bolinches and Esteban-Romero. Moreover,we extend a well-19 known theorem of Peng [Finite groups with pro-normal subgroups, Proc.Amer.Math.20Soc.20(1969)232–234]provingthatforalargeclassofgeneralizedFC-groupstheweak21transitivityofnormalityisequivalenttohavingfinitelymanymaximalpronormalizers22ofsubgroup