In this chapter we start the discussion of torsion-free groups. First, we deal with general properties along with the finite rank case, and delegate the in-depth theory of torsion-free groups of infinite rank to the next chapter.
After presenting the basic definitions and facts, we enter the study of balancedness, a stronger version of purity, which we have already met in the theory of torsion groups. Turning to the problem of direct decompositions, we start with the discussion of indecomposable groups; we do not restrict ourselves to the finite rank case as it seems more natural to deal with this important problem without rank restrictions. Concentrating on the finite rank case, the study of pathological direct decompositions is followed by positive results, the highlight being Lady’s theorem about the finiteness of non-isomorphic direct decompositions.
Other aspects of direct decompositions are also discussed, including quasi- and near-homomorphisms. Finite rank dualities will also be dealt with.