Finite Nilpotent Groups Whose Cyclic Subgroups are TI-Subgroups

A subgroup H of a group G is called a TI-subgroup if H-g boolean AND H = 1 or H for all g is an element of G; and H is called quasi TI if C-G(x) <= N-G(H) for all non-trivial elements x is an element of H. A group G is called (quasi CTI-group) CTI-group if every cyclic subgroup of G is a (quasi TI-subgroup) TI-subgroup. It is clear that TI subgroups are quasi TI. We first show that finite nilpotent quasi CTI-groups are CTI. In this paper, we classify all finite nilpotent CTI-groups.