Journal of Symbolic Logic

On properties of (weakly) small groups

Cédric Milliet

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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.

Article information

Source
J. Symbolic Logic, Volume 77, Issue 1 (2012), 94-110.

Dates
First available in Project Euclid: 20 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1327068693

Digital Object Identifier
doi:10.2178/jsl/1327068693

Mathematical Reviews number (MathSciNet)
MR2951631

Zentralblatt MATH identifier
1243.03052

Subjects
Primary: 03C45, 03C60, 20E45, 20E99, 20F18, 20F24

Keywords
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

Citation

Milliet, Cédric. On properties of (weakly) small groups. J. Symbolic Logic 77 (2012), no. 1, 94--110. doi:10.2178/jsl/1327068693. https://projecteuclid.org/euclid.jsl/1327068693


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