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Countable random $ p$-groups with prescribed Ulm-invariants

Authors: Manfred Droste and Rüdiger Göbel
Journal: Proc. Amer. Math. Soc. 139 (2011), 3203-3216
MSC (2000): Primary 20K10, 20K30; Secondary 60F20, 16W20
Published electronically: February 25, 2011
MathSciNet review: 2811276
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Abstract: In this paper we present a probabilistic construction of countable abelian $ p$-groups with prescribed Ulm-sequence. This result provides a different proof for the existence theorem of abelian $ p$-groups with any given countable Ulm-sequence due to Ulm, which is sometimes called Zippin's theorem. The basic idea, applying probabilistic arguments, comes from a result by Erdős and Rényi. They gave an amazing probabilistic construction of countable graphs which, with probability $ 1$, produces the universal homogeneous graph, therefore also called the random graph. P. J. Cameron says about this in his book Oligomorphic Permutation Groups [Cambridge University Press, 1990]: In 1963, Erdős and Rényi proved the following paradoxical result. ...It is my contention that mathematics is unique among academic pursuits in that such an apparently outrageous claim can be made completely convincing by a short argument. The algebraic tool in the present paper needs methods developed in the 1970s, the theory of valuated abelian $ p$-groups. Valuated abelian $ p$-groups are natural generalizations of abelian $ p$-groups with the height valuation, investigated in detail by F. Richman and E. Walker, and others. We have to establish extensions of finite valuated abelian $ p$-groups dominated by a given Ulm-sequence. Probabilistic results of a similar nature have been established by A. Blass and G. Braun, and by M. Droste and D. Kuske.

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Additional Information

Manfred Droste
Affiliation: Institute of Computer Science, Universität Leipzig, PF 100920, 04009 Leipzig, Germany

Rüdiger Göbel
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany

Keywords: Ulm’s theorem, abelian $p$-groups, valuated group, amalgamation
Received by editor(s): March 13, 2010
Received by editor(s) in revised form: July 9, 2010, and August 22, 2010
Published electronically: February 25, 2011
Additional Notes: The authors are supported by the project No. 963-98.6/2007 of the German-Israeli Foundation for Scientific Research & Development and by a project AOBJ 548025 of the Deutsche Forschungsgemeinschaft.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2011 American Mathematical Society

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