Products of minimal abelian groups
type: Item
instance of: scholarly article
title: Products of minimal abelian groups
zbMATH Open document ID: 0685.22001
DOI: 10.1007/BF02570894
author: Dikran Dikranjan
author: Dmitri Shakhmatov
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183791
review text: A Hausdorff topological group G is said to be minimal if every continuous group isomorphism \(G\to H\) onto a Hausdorff topological group H is a homeomorphism. A minimal group G is said to be perfectly minimal if the product \(G\times H\) is minimal for every minimal group H. \textit{D. Doitchinov} [Bull. Sci. Math., II. Sér. 96, 59-64 (1972; Zbl 0236.22002)] gave examples of non-compact minimal groups and non- perfectly-minimal minimal groups. \textit{L. Stoyanov} [Math. Educ. Math., Proc. 11th Spring Conf., Sunny Beach/Bulg. 1982, 287-294 (1982; Zbl 0528.22001)] characterized the perfectly minimal abelian groups and the minimal groups for which every power is minimal, showing in particular that the latter groups are precisely the minimal abelian groups G such that \(G^ c\) is minimal. Denote by \(\kappa\) the smallest cardinal \(\tau\) such that for every minimal abelian group G the minimality of \(G^{\tau}\) yields the minimality of all powers of G, then \(\aleph_ 0\leq \kappa \leq c\) [Stoyanov, loc. cit.]. The aim of the paper under review is to establish that \(\kappa\geq \gamma\), where \(\gamma\) is the smallest cardinal \(\tau\) such that \(2^{\tau}>c\). This yields in particular \(\kappa >\aleph_ 0\), and under CH or MA \(\kappa =c\). The proof is based on minimality and perfect minimality criteria for products of minimal abelian groups based on p-adic techniques and announced by the first author [C. R. Acad. Sci., Paris, Sér. I 299, 303-306 (1984; Zbl 0565.22002)]. The paper provides proofs and a lot of applications of these criteria, in particular a lot of examples are given - classes of minimal abelian groups with countable or uncountable \(\kappa\), minimal groups G such that \(G^ n\) is minimal for some natural n but \(G^{n+1}\) is not minimal etc. [See also the monograph of \textit{I. Prodanov}, \textit{L. Stoyanov} and the first author ``Topological Groups'' (Pure Appl. Math. 130, 1989; Zbl 0687.22001).]
reviewed by: Dikran Dikranjan
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 20K45
Mathematics Subject Classification ID: 54A10
zbMATH DE Number: 4122243
zbMATH Keywords: Hausdorff topological group
zbMATH Keywords: non-compact minimal groups
zbMATH Keywords: non-perfectly- minimal minimal groups
zbMATH Keywords: perfectly minimal abelian groups
zbMATH Keywords: minimal abelian groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4122243