Direct Sums of Cyclic Groups
The study of important classes of abelian groups begins in this chapter. Not counting the finite and finitely generated groups, the class of direct sums of cyclic groups is perhaps the best understood class.
We give a fairly detailed account of free abelian groups, and discuss the presentation of groups via generators and defining relations. Several sections are devoted to direct sums of cyclic groups (called \(\Sigma \)-cyclic groups); these groups share most useful properties, and can easily be characterized by cardinal invariants. We present a few criteria for such groups, and establish several remarkable results, e.g. Kulikov’s theorem that passage to subgroups preserves \(\Sigma \)-cyclicity. We draw attention to the method of smooth chains, which became the most important tool in the theory, and provides basic machinery for several results to come.
We shall cover some of the aspects of almost free groups, but shall not pursue their theory farther, due to the sophisticated set-theoretical arguments required.
In this chapter, in a number of proofs we have to use purity, so readers should be familiar with the fundamental results on pure subgroups (in Chapter 5) before studying the second part of this chapter.