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Proceedings of the American Mathematical Society

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Quasi-minimal abelian groups


Authors: B. Goldsmith, S. Óhógáin and S. Wallutis
Journal: Proc. Amer. Math. Soc. 132 (2004), 2185-2195
MSC (2000): Primary 20K99
DOI: https://doi.org/10.1090/S0002-9939-04-07065-0
Published electronically: March 24, 2004
MathSciNet review: 2052393
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Abstract: An abelian group $G$ is said to be quasi-minimal (purely quasi-minimal, directly quasi-minimal) if it is isomorphic to all its subgroups (pure subgroups, direct summands, respectively) of the same cardinality as $G$. Obviously quasi-minimality implies pure quasi-minimality which in turn implies direct quasi-minimality, but we show that neither converse implication holds. We obtain a complete characterisation of quasi-minimal groups. In the purely quasi-minimal case, assuming GCH, a complete characterisation is also established. An independence result is proved for directly quasi-minimal groups.


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Additional Information

B. Goldsmith
Affiliation: School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
Email: brendan.goldsmith@dit.ie

S. Óhógáin
Affiliation: School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
Email: johog@maths.tcd.ie

S. Wallutis
Affiliation: Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany
Email: simone.wallutis@uni-essen.de

DOI: https://doi.org/10.1090/S0002-9939-04-07065-0
Received by editor(s): April 17, 2002
Received by editor(s) in revised form: September 25, 2002
Published electronically: March 24, 2004
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2004 American Mathematical Society