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On properties of (weakly) small groups

Published online by Cambridge University Press:  12 March 2014

Cédric Milliet
Cédric Milliet*
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 CNRS 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
*
Université Galatasaray, Faculté de Sciences et de Lettres, Département de Mathématiques, Çiraǧan Caddesi 36, 34357 Ortaköy, Istamboul, Turquie, E-mail: [email protected]
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Abstract

A group is small if it has only countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has only countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable 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. First corollary: a weakly small group with simple theory has an infinite definable finite-by-abelian subgroup. Secondly, in a group with simple theory, a solvable group A of derived length n is contained in an A-definable almost solvable group of class at most 2n – 1.

Keywords

Small groupweakly small groupCantor-Bendixson ranklocal chain conditioninfinite abelian subgroupgroup in a simple theoryinfinite finite-by-abelian subgroupnilpotent group
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 94 - 110
Copyright
Copyright © Association for Symbolic Logic 2012

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