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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus

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Volume 16, Issue 2


Degeneration and orbits of tuples and subgroups in an abelian group

Wesley Calvert
  • Department of Mathematics, Mail Code 4408, 1245 Lincoln Drive, Southern Illinois University, Carbondale, Illinois 62901, USA
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/ Kunal Dutta / Amritanshu Prasad
Published Online: 2013-03-01 | DOI: https://doi.org/10.1515/jgt-2012-0038


A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably-generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained.

About the article

Received: 2012-08-26

Published Online: 2013-03-01

Published in Print: 2013-03-01

Citation Information: Journal of Group Theory, Volume 16, Issue 2, Pages 221–233, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgt-2012-0038.

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© 2013 by Walter de Gruyter Berlin Boston.Get Permission

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