Finite Nilpotent Groups Whose Cyclic Subgroups are TI-Subgroups

A subgroup H of a group G is called a TI-subgroup if Hg∩H=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^g\cap H=1$$\end{document} or H for all g∈G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in G$$\end{document}; and H is called quasi TI if CG(x)≤NG(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_G(x)\le \mathcal {N}_G(H)$$\end{document} for all non-trivial elements x∈H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in H$$\end{document}. 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.