A subgroup H of a finite group G is called a TI-subgroup, if H?H^x=? or H?H^x=H for all ?G . A group G is called a TI-group, if all of whose subgroups are TI-subgroups. The concept of TI-subgroup is fundamental role in the study of finite groups. In this thesis, we consider QH-groups, i.e. groups whose all metacyclic subgroups are TI-subgroups. We show that all QH-groups are solvable and classify QH-groups completely. Then we consider finite nilpotent groups whose cyclic subgroups are TI–subgroups. A subgroup H of G is called a QTI–subgroup if C_G (x)?N_G (H) for all ??x?H , where C_G (x) and N_G (H) centralizer x in G and normalizer H in G , respectively. A group G is called a CTI–group, if every cyclic subgroup is TI–subgroup. We show that every finite nilpotent group is quasi CTI if and only if is CTI.
