Almost Completely Decomposable Groups and Integral Linear Algebra

Ž . An almost completely decomposable group X is a finite essential abelian extension of a completely decomposable group A of finite rank. Almost completely decomposable groups appeared early in studies of torsion-free abelian groups, but only as examples, usually exhibiting pathological dew x compositions. In 1974 Lady Lad74 initiated a systematic theory of such groups based on the fundamental concept of regulating subgroup. The regulating subgroups can be defined as the completely decomposable subgroups of least index in an almost completely decomposable group. Details on the subsequent developments can be found in the survey article w x Mad95 . Also see the remarks preceding Theorem 5.1. Almost completely decomposable groups are easily written down. In fact, one traditionally starts with a completely decomposable group A inside a divisible hull Q A,