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Supersimple \(\omega\)-categorical groups and theories. (English) Zbl 0965.03050

The main result is that any \(\omega\)-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. A key ingredient is a proof (using the Schlichting/Bergman-Lenstra result in group theory) that if \(G\) is an \(\omega\)-categorical finite-by-abelian-by-finite group of finite SU-rank, then any definable subgroup of \(G\) is commensurable with one definable over the algebraic closure of \(\emptyset\). This is used to show both that any \(\omega\)-categorical group of finite SU-rank is finite-by-abelian-by-finite, and that any \(\omega\)-categorical supersimple group has finite SU-rank. The proof also uses the fact that any \(\omega\)-categorical group without the strict order property is nilpotent-by-finite. The paper also contains a proof that if \(T\) is any \(\omega\)-categorical simple theory and \(p\) is a CM-trivial regular type defined over \(\emptyset\), then \(p\) is non-orthogonal to a type of SU-rank 1. It follows that any \(\omega\)-categorical supersimple CM-trivial theory has finite SU-rank.

MSC:

03C60 Model-theoretic algebra
03C35 Categoricity and completeness of theories
03C45 Classification theory, stability, and related concepts in model theory
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