A transitively normal subgroup of a group $G$ is one that is normal in every subgroup in which it is subnormal. This concept is obviously related to the transitivity of normality because the latter holds in every subgroup of a group $G$ if and only if every subgroup of $G$ is transitively normal. In this paper we describe the structure of a group whose cyclic subgroups (or a part of them) are transitively normal.
"On the influence of transitively normal subgroups on the structure of some infinite groups." Publ. Mat. 57 (1) 83 - 106, 2013.