Groups whose non-normal subgroups have a transitive normality relation (Q1864698)

The structure of T-groups (or groups for which the normality relation in \(G\) is transitive), were described by Gaschütz for finite soluble groups and by Robinson for infinite soluble groups. Later, Romalis and Sesekin investigated the meta-Hamiltonian groups or groups whose non-normal subgroups are Abelian. The class of minimal-non-T groups and the \(\mathcal X\)-groups or groups in which every non-normal subgroup is a T-group are investigated. In this paper, the soluble \(\mathcal X\)-groups are studied. It is proved that they can be periodic or meta-Hamiltonian. Also, it is obtained that the periodic soluble \(\mathcal X\)-groups have derived length at most 4.