Solvability in groups with a chain condition on uniformly definable subgroups

TL;DR

This paper proves a structure theorem for certain periodic locally soluble groups with chain conditions, showing they are soluble, and offers a model-theoretic proof of a related classical result.

Contribution

It provides a new, simpler proof that periodic locally soluble groups with chain conditions are soluble, using stable group theory techniques.

Findings

Groups satisfying the chain condition are soluble.

The proof simplifies existing results on periodic locally soluble groups.

It connects model theory with classical group theory results.

Abstract

We prove a structure theorem for periodic locally soluble groups satisfying a chain condition on intersections of relatively uniformly definable subgroups using results from the theory of stable groups. The result in particular shows that these groups are soluble, thus giving a model-theoretic and much simpler proof of a special case of a theorem of Bryant and Hartley on the solubility of periodic locally soluble groups satisfying the minimal condition on centralizers.