Rigid abelian groups and the probabilistic method

Details Rigid abelian groups and the probabilistic method Rigid abelian groups and the probabilistic method

2011-07-12

arxiv.org/abs/1107.2325v1

Mathematics

Group Theory

12 pages, submitted to the special volume of Contemporary Mathematics for the proceedings of the conference Group and Model Th

Scientific paper

The construction of torsion-free abelian groups with prescribed endomorphism rings starting with Corner's seminal work is a well-studied subject in the theory of abelian groups. Usually these construction work by adding elements from a (topological) completion in order to get rid of (kill) unwanted homomorphisms. The critical part is to actually prove that every unwanted homomorphism can be killed by adding a suitable element. We will demonstrate that some of those constructions can be significantly simplified by choosing the elements at random. As a result, the endomorphism ring will be almost surely prescribed, i.e., with probability one.

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