Bounded essential extensions of completely decomposable groups
This paper concerns bcd groups, i.e. torsion-free Abelian groups of infinite rank which contain a completely decomposable subgroup of bounded index. These bcd groups are natural generalizations of the well-known acd groups, i.e. groups of the same description with infinite replaced by finite. One of the most significant concepts in the study of acd groups is that of regulating subgroup, which is defined in two equivalent ways: as a completely decomposable subgroup of minimal index, or as a direct sum of homogeneous completely decomposable subgroups obtained from Butler decompositions of the type subgroups. Now it turns out that in the infinite rank case, these definitions make sense but are no longer equivalent, and only the second one is useful. The authors show that all bcd groups have regulating subgroups in this second sense, but the quotient groups are not as controlled as in the acd case. They reveal the most important properties of regulating subgroups and their quotients, find numerous interesting examples and present several unsolved problems, of which the most important seems to be whether an infinite rank direct summand of a bcd group is a bcd group.
almost completely decomposable Abelian groups
torsion-free Abelian groups of infinite rank
completely decomposable subgroups of bounded index
scientific article; zbMATH DE number 1496010