Hall classes of groups

Abstract

In 1958, Philip Hall (Ill J Math 2:787–801, 1958) proved that if a group G has a nilpotent normal subgroup N such that G/N′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G/N'$$\\end{document} is nilpotent, then G is nilpotent. The scope of Hall’s nilpotency criterion is not restricted to group theory, and in fact similar statements hold for Lie algebras and more generally for algebraically coherent semiabelian categories (see Chao in Math Z 103:40–42, 1968; Gray in Adv Math 349:911–919, 2019; Stitzinger in Ill J Math 22:499–505, 1978). We say that a group class 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 a Hall class if it contains every group G admitting a nilpotent normal subgroup N such that G/N′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G/N'$$\\end{document} belongs to 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}. Thus, Hall’s nilpotency criterion just asserts that nilpotent groups form a Hall class. Many other relevant classes of groups have been proved to be Hall classes; for example, Plotkin (Sov Math Dokl 2:471–474, 1961) and Robinson (Math Z 107:225–231, 1968) proved that locally nilpotent groups and hypercentral groups form Hall classes. Note that these generalizations also hold if groups are replaced by other algebraic structures, for example Lie algebras (see Stitzinger in Ill J Math 22:499–505, 1978). The aim of this paper is to develop a general theory of Hall classes of groups, that could later be reasonably extended to Lie algebras. Among other results, we prove that many natural types of generalized nilpotent groups form Hall classes, and we give examples showing in particular that the class of groups having a finite term in the lower central series is not a Hall class, even if we restrict to the universe of linear groups.