A note on commutator subgroups in groups of large cardinality

被引：0
作者：
Maria De Falco
Francesco de Giovanni
Carmela Musella
机构：
[1] Università di Napoli Federico II,Dipartimento di Matematica e Applicazioni
[2] Complesso Universitario Monte S. Angelo,undefined
来源：
Monatshefte für Mathematik | 2020年 / 191卷
关键词：
Uncountable group; Commutator subgroup; Normality; 20F14;
D O I：
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摘要：
If G is an uncountable group of regular 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}, we shall denote by LLℵ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {L}L}_\aleph (G)$$\end{document} the set of all subgroups of G 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}. The aim of this paper is to describe the behaviour of groups G for which the set Cℵ(G)={X′|X∈LLℵ(G)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {C}}}_\aleph (G)=\{ X'\;|\; X\in {\mathfrak {L}L}_\aleph (G)\}$$\end{document} is finite, at least when G is locally graded and has no simple sections 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}. Among other results, it is proved that such a group has a finite commutator subgroup, provided that it contains an abelian subgroup 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}.
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页码：249 / 256
页数：7
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