On the minimality of powers

of minimal -bounded abelian groups

Authors:
Dikran Dikranjan and Alberto Tonolo

Journal:
Proc. Amer. Math. Soc. **127** (1999), 3101-3110

MSC (1991):
Primary 54F45, 54D05, 54D30; Secondary 22A05, 22D05, 54D25

DOI:
https://doi.org/10.1090/S0002-9939-99-04834-0

Published electronically:
April 23, 1999

MathSciNet review:
1600132

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Abstract | References | Similar Articles | Additional Information

Abstract: We describe the structure of totally disconnected minimal -

bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the -adic integers . In this case the description is obtained by means of a functorial correspondence, based on Pontryagin duality, between topological and linearly topologized groups introduced by Tonolo. As an application we answer the question (posed in *Pseudocompact and countably compact abelian groups: Cartesian products and minimality*, Trans. Amer. Math. Soc. **335** (1993), 775-790) when arbitrary powers of minimal -bounded abelian groups are minimal. We prove that the positive answer to this question is equivalent to non-existence of measurable cardinals.

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Additional Information

**Dikran Dikranjan**

Affiliation:
Dipartimento di Matematica e Informatica, Udine University, via delle Scienze 206 (località Rizzi), 33100 Udine, Italy

Email:
dikranja@dimi.uniud.it

**Alberto Tonolo**

Affiliation:
Dipartimento di Matematica Pura ed Applicata, Padova University, Via Belzoni 7, 35131 Padova, Italy

Email:
tonolo@math.unipd.it

DOI:
https://doi.org/10.1090/S0002-9939-99-04834-0

Keywords:
Totally disconnected group,
connected group,
countably compact group,
$\omega $-bounded group,
minimal group,
measurable cardinal

Received by editor(s):
March 25, 1997

Received by editor(s) in revised form:
December 13, 1997

Published electronically:
April 23, 1999

Communicated by:
Alan Dow

Article copyright:
© Copyright 1999
American Mathematical Society