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 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 . 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