Abelian Groups - László Fuchs [Springer]

Chapter 13
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.
1 Direct Decompositions of Infinite Rank Groups
In our discussions of torsion-free groups of infinite rank, the first program is to
study their direct decompositions. One can say quite a bit about their strange
decompositions already in the countable rank case, and we will concentrate on this
cardinality.
In contrast to the finite rank case, countable rank groups need not be direct sums
of indecomposable groups. Some rather paradoxical behavior will be shown in the
theorems that follow. In view of our experience with finite rank groups, it should
not be a surprise that nearly everything conceivable can occur in the countable case
as well.
Decompositions into Indecomposable Summands First we consider the case
when the groups are better behaved, and admit decompositions into indecomposable
summands. The first question we take up is concerned with the number of
summands.
Theorem 1.1 (Corner [1]). There exists a torsion-free group A that has two
decompositions
A D B ˚ C D M n2Z
E n ;
© Springer International Publishing Switzerland 2015
L. Fuchs, Abelian Groups, Springer Monographs in Mathematics,
DOI 10.1007/978-3-319-19422-6_13
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