Finite Nilpotent Groups Whose Cyclic Subgroups Are TI-Subgroups

A subgroup H of a group G is called a TI-subgroup if \(H^g\cap H=1\) or H for all \(g\in G\); and H is called quasi TI if \(\mathcal {C}_G(x)\le \mathcal {N}_G(H)\) for all non-trivial elements \(x\in H\). 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.