On properties of (weakly) small groups

TL;DR

This paper explores properties of weakly small groups, showing how their subgroups behave under algebraic closure conditions, and establishing the existence of certain definable subgroups in groups with simple theory.

Contribution

It introduces new structural results about weakly small groups, including conditions for subgroups and the existence of definable abelian and solvable subgroups in simple theories.

Findings

Weakly small groups have subgroups satisfying descending chain conditions.

Infinite weakly small groups contain infinite abelian subgroups.

In groups with simple theory, certain sets are contained in definable finite-by-abelian subgroups.

Abstract

A group is small if it has 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 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 subgoup. Secondly, in a group with simple theory, a normal solvable group A of derived length n is contained in an A-definable almost solvable group of class n.