GROUPS WHOSE PROPER SUBGROUPS OF INFINITE RANK ARE FINITE-BY-HYPERCENTRAL OR HYPERCENTRAL-BY-FINITE

A group G is said to be of finite rank r if every finitely generated subgroup of G can be generated by at most r elements, and r is the least positive integer with a such property. If there is no such r, then the group G is said to be of infinite rank. In the present paper, it is proved that if G is an X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{X}$$\end{document}-group of infinite rank whose proper subgroups of infinite rank are finite-by-hypercentral (respectively, hypercentral-by-finite), then all proper subgroups of G are finite-by-hypercentral (respectively, hypercentral-by-finite), where X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{X}$$\end{document} is the class defined by N.S. Černikov as the closure of the class of periodic locally graded groups by the closure operations P´\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\acute{P}}$$\end{document}, P`\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\grave{P}}$$\end{document} and L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{L}$$\end{document}.