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Preprint Number 226

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226. Cédric Milliet
A property of small groups
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Submission date: 18 January 2010.

Abstract:

A group is small if it has countably many pure n-types for each integer n. It is shown that in a small group, subgroups which are definable with parameters in a finitely generated algebraic closure satisfy local descending chain conditions. An infinite small group has an infinite abelian subgroup, which may not be definable. A nilpotent small group is the central product of a definable divisible group with one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. As a corollary, a group with small and simple theory has an infinite definable finite-by-abelian subgroup.

Mathematics Subject Classification: 03C45,03C60,20E45,20F18,20F24.

Keywords and phrases: Small group, weakly small group, Cantor-Bendixson rank, local chain condition, infinite abelian subgroup, group in a simple theory, infinite finite-by-abelian subgroup, nilpotent group.

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