Torsion-Free Groups of Infinite Rank

Abstract

This chapter continues the theme of torsion-free groups, this time for the infinite rank case. There is no shortage of relevant results.

After a short discussion of direct decompositions of countable torsion-free groups, we enter the study of slender groups which display remarkable phenomena. We provide the main results on this class of groups. Much can be said about separable and vector groups. These seem theoretically close to completely decomposable groups, but are less tractable, and so more challenging. The measurable case is quite interesting.

The theory of torsion-free groups would not be satisfactorily dealt with without the discussion of the Whitehead problem. For a quarter of century this was the main open problem in abelian groups. We will give a detailed proof of its undecidability, mimicking Shelah’s epoch-making solution. We show that the answers are different in the constructible universe, and in a model of set theory with Martin’s Axiom and the denial of CH.