Abelian Groups - László Fuchs [Springer]

Chapter 14
Butler Groups
Abstract The theory of Butler groups is one of the most elaborate branches of abelian groups.
We devote a whole chapter to its study. This may seem excessive, since Butler groups are very
special, but the fact that the results and the methods provide more than a superficial glimpse into
a fascinating theory (the most extensive one today on torsion-free groups of arbitrary rank) is a
compelling reason for including a broader discussion.
In the finite rank case, Butler groups are torsion-free generated by a finite number of rank 1
groups. It is very tempting to think that a kind of finite generation makes them susceptible to a
satisfactory classification; however, so far no comprehensive theory has emerged, though many
encouraging results are available. We prove the basic results on them, but we are unable to dig
deeply into the theory without getting involved in overcomplicated details. An important part of the
theory aims at finding more tractable classes of finite rank Butler groups. The fast developing, very
successful theory of almost completely decomposable groups is well documented in Mader[Ma].
We investigate more thoroughly Butler groups of large cardinalities; their theory has gained
considerable popularity in the last quarter of the twentieth century, and even today they remain
under intense scrutiny. It is an illuminating experience to see the effect of set theory on their
algebraic structure.
1 Finite Rank Butler Groups
The only class of finite rank torsion-free groups that is larger than the class of
completely decomposable groups and has a well developed theory is the class of
Butler groups. This class is more manageable than finite rank torsion-free groups
in general, but still general enough to generate a variety of challenging problems.
This is still a fertile research area. The theory originates from Butler [1] and Bican
[3], using different approaches. The fundamental facts on this class of groups are
due to them.
Basic Properties of Butler Groups After these prefatory remarks, we now state
the precise definition. A torsion-free group of finite rank is called a Butler group if
it is a pure subgroup of a completely decomposable group (of finite rank).
Evidently, all completely decomposable groups of finite rank, in particular, all
rank 1 groups, are Butler groups. The main interest lies, of course, in Butler groups
that fail to be completely decomposable.
Example 1.1. Let A be the direct sum: A D A 1 ˚ A 2 ˚ A 3 where each A i is torsion-free of rank 1
such that they contain elements a i 2 A i of characteristics .1; 1;0;0;:::/;.1;0;1;0;0;:::/,
© Springer International Publishing Switzerland 2015
L. Fuchs, Abelian Groups, Springer Monographs in Mathematics,
DOI 10.1007/978-3-319-19422-6_14
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