Groups in which normality is a weakly transitive relation.

We study groups in which normality is a weakly transitive relation, giving an extension of in Theorem A [On finite T-groups, J. Aust. Math. Soc. 75 (2003) 181-191] due to Ballester-Bolinches and Esteban-Romero pointing out the relations between these groups and those in which all subgroups are almost pronormal. Moreover, we extend a well-known theorem of Peng [Finite groups with pro-normal subgroups, Proc. Amer. Math. Soc. 20 (1969) 232-234] proving that for a large class of generalized FC-groups the weak transitivity of normality is equivalent to having finitely many maximal pronormalizers of subgroups. [ABSTRACT FROM AUTHOR]