A note on uncountable groups with modular subgroup lattice

The aim of this paper is to investigate the behaviour of uncountable groups of cardinality ℵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph}$$\end{document} in which all proper subgroups of cardinality ℵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph}$$\end{document} have modular subgroup lattice. It is proved here that the lattice of subgroups of such a group G is modular, provided that G has no infinite simple homomorphic images of cardinality ℵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph}$$\end{document}. A corresponding result for groups whose proper subgroups of large cardinality are quasihamiltonian is also proved.