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.