Finite Nilpotent Groups Whose Cyclic Subgroups are TI-Subgroups

Abstract

A subgroup H of a group G is called a TI-subgroup if Hg∩ H= 1 or H for all g∈ G; and H is called quasi TI if CG(x) ≤ NG(H) for all non-trivial elements x∈ 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. © 2015, Malaysian Mathematical Sciences Society and Universiti Sains Malaysia.