Direct decompositions of torsion-free Abelian groups of finite rank

Abstract

It is proved that ifr1,r2, ...,rs;l1,l2, ...,lt are the ranks of the indecomposable summands of two direct decompositions of a torsion-free Abelian group of finite rank and if s0 is the number of units among the numbers ri, while t0 is the number of units among the numbers lj, thenri ⩽n - t0,lj⩽n−s0 for all i, j. Moreover, if for some i we have ri=n−t0, then among the lj's only one term is different from 1 and it is equal to n−t0; similarly if lj=n−s0 for some j. In addition, a construction is presented, allowing to form, from several indecomposable groups, a new group, called a flower group, and it is proved that a flower group is indecomposable under natural restrictions on its defining parameters.