Butler groups of infinite rank Original Research Article

Butler groups are torsion-free abelian groups which — in the infinite rank case — can be defined in two different ways. One definition requires that all the balanced extensions of torsion groups by them are splitting, while the other stipulates that they admit continuous transfinite chains (with finite rank factors) of so-called decent subgroups. This paper is devoted to the three major questions for Butler groups of infinite rank: Are the two definitions equivalent? Are balanced subgroups of completely decomposable torsion-free groups always Butler groups? Which pure subgroups of Butler groups are again Butler groups? In attacking these problems, a new approach is used by utilizing aleph, Hebrew0-prebalanced chains and relative balanced-projective resolutions introduced by Bican and Fuchs [5]. A noteworthy feature is that no additional set-theoretical hypotheses are needed.