Abelian Groups - László Fuchs [Springer]

Chapter 12
Torsion-Free Groups
Abstract 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 nearhomomorphisms.
Finite rank dualities will also be dealt with.
1 Characteristic and Type: Finite Rank Groups
In this section, all groups are assumed to be torsion-free, unless stated otherwise.
p 1 ; p 2 ;:::;p n ;:::will denote the sequence of prime numbers.
Every torsion-free group A is a subgroup of a Q-vector space V such that a
maximal independent set in A is a basis of V.Iffx i .i 2 I/g is a maximal independent
set in A, then every element of A depends on it, and therefore every element a 2 A
can be written uniquely as
a D r 1 x 1 CCr k x k
.r i 2 Q/:
Note that only certain combinations of rational coefficients are allowed, depending
on the group and the choice of the maximal independent set.
Characteristics The most typical features of elements in torsion-free groups are
crystallized in the concept corresponding to height. Let A be a torsion-free group and
a 2 A. Recall that, for a prime p, the largest integer k with p k ja, i.e.,forwhichthe
equation p k x D a is solvable in A, is called the p-height h p .a/ of a. Ifnosuch
maximal integer k exists, then we set h p .a/ D1. The sequence of p-heights,
.a/ D .h p1 .a/; h p2 .a/;:::;h pn .a/; : : :/;
© Springer International Publishing Switzerland 2015
L. Fuchs, Abelian Groups, Springer Monographs in Mathematics,
DOI 10.1007/978-3-319-19422-6_12
409